Solutions: (ix) ${{a}_{n}}=\left( 2n-3 \right)/6$ Given sequence whose, ${{a}_{n}}=\frac{2n-3}{6}$ To get the first five terms of the sequence we put $n=1,2,3,4,5.$ And, we get...
1. Write the first terms of each of the following sequences whose ${{n}^{th}}$ term are: (vii) ${{a}_{n}}={{n}^{2}}-n+1$ (viii) ${{a}_{n}}={{n}^{2}}-n+1$
An arithmetic progressions or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions: (vii) ${{a}_{n}}={{n}^{2}}-n+1$ The given...
1. Write the first terms of each of the following sequences whose ${{n}^{th}}$ term are: (v) ${{a}_{n}}={{\left( -1 \right)}^{n}}{{.2}^{n}}$ (vi) ${{a}_{n}}=n\left( n-2 \right)/2$
An arithmetic progressions or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions: (v) ${{a}_{n}}={{\left( -1...
27. Two arithmetic progressions have the same common difference. The difference between their ${{100}^{th}}$ terms is $100,$ what is the difference between their ${{1000}^{th}}$ terms?
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Let the two A.Ps be $A.{{P}_{1}}$ and...
Which term of the A.P. $3,10,17,$ …. will be 84 more than its ${{13}^{th}}$ term ?
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Given A.P. is 3, 10, 17.... Here, a = 3 d = 10 – 3 = 7 Let...
The 7th term of an A.P. is $32$ and its ${{13}^{th}}$ term is $62.$ Find the A.P. An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant.
Solution: Given, ${{a}_{7}}=32$ and ${{a}_{13}}=62$ From ${{a}_{n}}-{{a}_{k}}=\left( a+nd-d \right)-\left( a+kd-d \right)$ $=\left( n-k \right)d$ where n and k represents the nth and kth terms of an...
24. Find the arithmetic progression whose third term is $16$ and the seventh term exceeds its fifth term by $12.$
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Given, in an A.P ${{a}_{3}}=16$ and...
23. The eighth term of an A.P is half of its second term and the eleventh term exceeds one third of its fourth term by $1.$ Find the ${{15}^{th}}$ term.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Given, an A.P in which, ${{a}_{8}}=1/2\left(...
22. Find n if the given value of $x$ is the ${{n}^{th}}$ term of the given A.P.
(iii) $5{\scriptstyle{}^{1}/{}_{2}},11,16{\scriptstyle{}^{1}/{}_{2}},22,....;x=550$ (iv) $1,21/11,31/11,41/11,...;x=171/11$ An arithmetic progression or arithmetic sequence is a number’s sequence...
22. Find n if the given value of $x$ is the ${{n}^{th}}$ term of the given A.P.
(i) $25,50,75,100,;x=1000$ (ii) $-1,-3,-5,-7,...;x=-151$ An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant....
21. Write the expression ${{a}_{n}}-{{a}_{k}}$ for the A.P. $a,a+d,a+2d,$ …..
Hence, find the common difference of the A.P. for which (iii) $20{}^{th}$ term is $10$ more than the ${{18}^{th}}$ term. An arithmetic progression or arithmetic sequence is a number’s sequence such...
21. Write the expression ${{a}_{n}}-{{a}_{k}}$ for the A.P. $a,a+d,a+2d,$ …..
Hence, find the common difference of the A.P. for which (i) ${{11}^{th}}$ term is $5$ and ${{13}^{th}}$ term is $79.$ (ii) ${{a}_{10}}-{{a}_{5}}=200$ An arithmetic progression or...
20. Find ${{a}_{30}}-{{a}_{20}}$ for the A.P.
(i) $-9,-14,-19,-24$ (ii) $a,a+d,a+2d,a+3d,$ …… An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution:...
19. The first term of an A.P. is $5$ and its ${{100}^{th}}$ term is $-292.$ Find the ${{50}^{th}}$ term of this A.P.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Given, an A.P whose $a=5$ and ${{a}_{100}}=-292$...
18. The sum of ${{4}^{th}}$ and ${{8}^{th}}$ terms of an A.P. is $24$ and the sum of the ${{6}^{th}}$ and ${{10}^{th}}$ terms is $34.$ Find the first term and the common difference of the A.P.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Given, in an A.P The sum of...
17. An A.P. consists of 60 terms. If the first and the last terms be $7$ and $125$ respectively, find ${{32}^{nd}}$ term.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Given, an A.P of $60$ terms And, $a=7$ and...
16. How many numbers of two digit are divisible by $3$?
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: The first 2 digit number divisible by $3$ is...
15. Find the second term and the ${{n}^{th}}$ term of an A.P. whose ${{6}^{th}}$ term is $12$ and the ${{8}^{th}}$ term is $22.$
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Given, in an A.P ${{a}_{6}}=12$ and...
14. The ${{4}^{th}}$ term of an A.P. is three times the first and the ${{7}^{th}}$ term exceeds twice the third term by $1.$ Find the first term and the common difference.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Let’s consider the first term and the common...
13. Find the ${{12}^{th}}$ term from the end of the following arithmetic progressions:
(iii) $1,4,7,10,$ … $,88$ An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: (iii) Given A.P...
13. Find the ${{12}^{th}}$ term from the end of the following arithmetic progressions:
(i) $3,5,7,9,$ …. $201$ (ii) $3,8,13,$ … $,253$ An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution:...
12. If the nth term of the A.P. $9,7,5,$…. is same as the nth term of the A.P. $15,12,9,$ … find n.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Given, $A.{{P}_{1}}=9,7,5,$…. and $A.{{P}_{2}}=15,12,9,$…...
11. The ${{26}^{th}}$, ${{11}^{th}}$ and the last term of an A.P. are $0,3$ and $-1/5,$ respectively. Find the common difference and the number of terms.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Given, ${{a}_{26}}=0,$${{a}_{11}}=3$ ,...
3. The general term of a sequence is given by an = -4n + 15. Is the sequence an A.P.? If so, find its 15th term and the common difference.
Solution: Given, ${{a}_{{{n}_{{}}}}}=-4n+15$ Now putting $n = 1, 2, 3, 4$ we get, ${{a}_{1}}=-4[1]+15=-4+15=11$ ${{a}_{2}}=-4[2]+15=-8+15=7$ ${{a}_{3}}=-4[3]+15=-12+15=3$ ...
2. Show that the sequence defined by an = 3n2 – 5 is not an A.P.
Solution: Given, ${{a}_{n}}=3{{n}_{2}}-5$ Now putting $n = 1, 2,3,4$ we get,...
1. Show that the sequence defined by an = 5n – 7 is an A.P., find its common difference.
Solution: Given, $an = 5n – 7$ Now putting $n = 1, 2, 3, 4$ we get, ${{a}_{1}}=5[1]=5-7=-2$$$ ${{a}_{2=}}5[2]-7=10-7=3$ ${{a}_{3}}=5[3]-7=15-7=8$ ${{a}_{4}}=5[4]-7=20-7=13$ We can see that,...
10. In a certain A.P. the ${{24}^{th}}$ term is twice the ${{10}^{ht}}$ term. Prove that the ${{72}^{nd}}$ term is twice the ${{34}^{th}}$ term.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions: Given, ${{24}^{th}}$ term is twice the...
9. The ${{10}^{th}}$ and ${{18}^{th}}$ terms of an A.P. are $41$ and $73$ respectively. Find ${{26}^{th}}$ term.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution: Given, ${{A}_{10}}=41$ and ${{a}_{18}}=73$ We...
8. If $10$ times the ${{10}^{th}}$ term of an A.P. is equal to $15$ times the ${{15}^{th}}$ term, show that ${{25}^{th}}$ term of the A.P. is zero.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions: Given, times the ${{10}^{th}}$ term...
7. If ${{9}^{th}}$ term of an A.P. is zero, prove its ${{29}^{th}}$ term is double the ${{19}^{th}}$ term.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions: Given, ${{a}_{9}}=0$ We know that,...
6. The ${{6}^{th}}$ and ${{17}^{th}}$ terms of an A.P. are $19$ and $41$ respectively, find the ${{40}^{th}}$ term.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions: Given, ${{a}_{6}}=19$ and ${{a}_{17}}=41$ We...
5. The first term of an A.P. is $5,$ the common difference is $3$ and the last term is $80$; find the number of terms.
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions: Given, $a=5$ and $d=3$ We know that,...
4. How many terms are there in the A.P.?
(iii) $7,13,19,$ …$,205$ (iv) $18,15{\scriptstyle{}^{1}/{}_{2}},13,$ ….$,-47$ An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive...
4. How many terms are there in the A.P.?
(i) $7,10,13,$ …..$,43$ (ii) $-1,-5/6,-2/3,-1/2,$ … $,10/3$ An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is...
3.(iii) Is $-150$ a term of the A.P. $11,8,5,2,$ …
An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions: (iii) Given, A.P. $11,8,5,2$ … Here, $a=11$ and...
3.(i) Is $68$ a term of the A.P. $7,10,13,$… ?
(ii) Is $302$ a term of the A.P. $3,8,13,$ …. ? An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions:...
2.
(v) Which term of the AP $121,117.113,$ … is its first negative term? An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms...
2.
(iii) Which term of the AP $4.9,14,$ …. is $254?$ (iv) Which term of the AP $21.42,63,84,$ … is $420?$ An arithmetic progression or arithmetic sequence is a number’s sequence such that the...
2.(i) Which term of the AP $3,8,13,$ …. is $248?$
(ii) Which term of the AP $84,80,76,$ … is $0?$ An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solution:...
1. Find:
(vii) ${{9}^{th}}$ term of the AP $3/4,5/4,7/4+9/4,$ ……….. An arithmetic progression or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant....
1. Find:
(v) ${{8}^{th}}$ term of the AP $11,104,91,78,$ …………… (vi) ${{11}^{th}}$ tenor of the AP $10.0,10.5,11.0,11.2,$ ………….. An arithmetic progression or arithmetic sequence is a number’s...
1. Find:
(iii) ${{n}^{th}}$ term of the AP $13,8,3,-2,$ ………. (iv) ${{10}^{th}}$ term of the AP $-40,-15,10,35,$ …………. An arithmetic progression or arithmetic sequence is a number’s sequence such that the...
1. Find:
(i) ${{10}^{th}}$ tent of the AP $1,4,7,10$…. (ii) ${{18}^{th}}$ term of the AP $\sqrt{2},3\sqrt{2},5\sqrt{2,}$……. An arithmetic progression or arithmetic sequence is a number’s sequence such...
2. Write the arithmetic progression when first term a and common difference $d$ are as follows:
(iii) $a=-1.5,d=-0.5$ An arithmetic progression is a number’s sequence such that the difference between the consecutive terms is constant. Formula for this is: $an=d\left( n-1 \right)+c,$ (iii)...
3. In which of the given situations, the sequence of numbers formed will form an A.P.?
(iii) Divya deposited Rs $1000$ at compound interest at the rate of $10%$ per annum. The amount at the end of first year, second year, third year, …, and so on. An arithmetic progression is a...
3. In which of the given situations, the sequence of numbers formed will form an A.P.?
(i) The cost of digging a well for the first metre is Rs $150$ and rises by Rs $20$ for each coming after metre.(ii) The amount of air present in the cylinder when a vacuum pump removes each time...
2. Write the arithmetic progression when first term a and common difference $d$ are as follows:
(i) $a=4,d=-3$ (ii) $a=-1,d={1}/{2}\;$ An arithmetic progression is a number’s sequence such that the difference between the consecutive terms is constant. Formula for this is: $an=d\left( n-1...
1. For the following arithmetic progressions write the first term a and the common difference $d$:
(iii) $0.3,0.55,0.80,1.05,...$ (iv) $-1.1,-3.1,-5.1,-7.1,...$ An arithmetic progression is a number’s sequence such that the difference between the consecutive terms is constant. Formula for this...
1. For the following arithmetic progressions write the first term a and the common difference $d$:
(i) $-5,-1,3,7,...$ (ii) ${1}/{5,{3}/{5}\;}\;,{5}/{5}\;,{7}/{5}\;,...$ An arithmetic progression is a number’s sequence such that the difference between the consecutive terms is constant. Formula...
In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato and other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line. A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?
[Hint: to pick up the first potato and the second potato, the total distance (in metres) run by a competitor is 2×5+2×(5+3)]
Solution: The distances between the bucket and the potatoes are 5, 8, 11, 14,..., which is in the form of AP. Given that the competitor's journey for gathering these potatoes is two times the...
200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed and how many logs are in the top row?
Solution: The numbers of logs in each row are in the form of an A.P. as 20, 19, 18,... For the provided A.P., The first term, a = 20 and the common...
A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, ……… as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take π = 22/7)
Solution: We are aware that, The perimeter of a semi-circle is πr Therefore, P1 = π(0.5) = π/2 cm P2 = π(1) = π cm P3 = π(1.5) = 3π/2 cm We can say that, P1, P2, P3 are...
In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of class I will plant 1 tree, a section of class II will plant 2 trees and so on till class XII. There are three sections of each class. How many trees will be planted by the students?
Solution: It is clear that the number of trees planted by students is in the form of A.P. 1, 2, 3, 4, 5………………..12 The first term, a = 1 The common difference, d = 2−1 = 1...
A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.
Solution: Let us assume that Rs. P be the cost of1st prize Then the cost of 2nd prize = Rs. P − 20 And the cost of 3rd prize = Rs. P − 40 We can see...
A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc., the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days.
Solution: The given penalties are in the form of an A.P. having the first term as 200 and the common difference as 50, as can be seen. As a result, a = 200...
Find the sum of the odd numbers between 0 and 50.
Solution: 1, 3, 5, 7, 9 … 49 are the odd numbers between 0 and 50. As a result, we can see that these odd numbers are in the form of A.P. Hence, The first term, a = 1 The common difference,...
Find the sum of first 15 multiples of 8.
Solution: 8, 16, 24, 32… are the multiples of 8 This series creates an A.P., with the first term being 8 and the common difference being 8. As a result, a = 8 d = 8 S15 = ? Using...
Find the sum of first 40 positive integers divisible by 6.
Solution: 6, 12, 18, 24 …. are the positive integers that are divisible by 6 This series creates an A.P., with the first term being 6 and the common difference being 6. a = 6 d = 6...
If the sum of the first n terms of an AP is 4n − n2, what is the first term (that is S1)? What is the sum of first two terms? What is the second term? Similarly find the 3rd, the10th and the nth terms.
Solution: Provided here that, Sn = 4n−n2 The first term, a = S1 = 4(1) − (1)2 = 4−1 = 3 The sum of first two terms = S2= 4(2)−(2)2 = 8−4 = 4 The second...
Show that a1, a2 … , an , … form an AP where an is defined as below
(i) an = 3+4n(ii) an = 9−5nAlso find the sum of the first 15 terms in each case. Solutions: (i) an = 3+4n a1 = 3+4(1) = 7 a2 = 3+4(2) = 3+8 = 11 a3 = 3+4(3) = 3+12 = 15...
If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.
Solution: Provided here that, S7 = 49 S17 = 289 We are aware that, Sum of n terms; Sn = n/2 [2a + (n – 1)d] As a result, S7= 7/2 [2a +(n -1)d]...
Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18, respectively.
Solution: Provided here that, The second term, a2 = 14 The third term, a3 = 18 The common difference, d = a3−a2 = 18−14 = 4 a2 = a+d 14 = a+4 a = 10 =...
Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
Solution: Provided here,The common difference, d = 7 22nd term, a22 = 149 The sum of the first 22 term, S22 = ? Using the nth term formula, an = a+(n−1)d...
The first and the last term of an AP are 17 and 350, respectively. If the common difference is 9, how many terms are there and what is their sum?
Solution: Provided that, The first term, a = 17 The last term, l = 350 The common difference, d = 9 If the A.P. has n terms, the formula for the last term can be expressed as; l =...
The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
Solution: Provided that, The first term, a = 5 The last term, l = 45 The sum of the AP be, Sn = 400 The sum of the AP formula, as we all know, is; Sn = n/2 (a+l) 400...
How many terms of the AP. 9, 17, 25 … must be taken to give a sum of 636?
Solutions: In the given AP. 9, 17, 25 … let there be n terms For the above A.P., First term, a = 9 The common difference, d = a2−a1 = 17−9 = 8 Say that the total of n terms...
In an AP
(i) Given a = 3, n = 8, S = 192, find d.(ii) Given l = 28, S = 144 and there are total 9 terms. Find a. Solutions: (i) Provided that, The first term, a = 3, The number of...
In an AP
(i) Given a = 8, an = 62, Sn = 210, find n and d.(ii) Given an = 4, d = 2, Sn = − 14, find n and a. Solutions: (i) Provided that, a = 8, an = 62, Sn = 210 In AP,...
In an AP(i) Given d = 5, S9 = 75, find a and a9.
(ii) Given a = 2, d = 8, Sn = 90, find n and an.
Solutions: (i) Provided that, d = 5, S9 = 75 In AP, the sum of n terms is, Sn = n/2 [2a +(n -1)d] As a result, the sum of the first nine terms is;...
In an AP
(i) Given a12 = 37, d = 3, find a and S12.(i) Given a3 = 15, S10 = 125, find d and a10. Solutions: (i) Provided that, a12 = 37, d = 3 The formula for the nth term in an AP,...
In an AP
(i) Given a = 5, d = 3, an = 50, find n and Sn.
(ii) Given a = 7, a13 = 35, find d and S13.
Solutions: (i) Provided that, a = 5, d = 3, an = 50 The formula for the nth term in an AP, as we know, is an = a +(n −1)d, Putting the given...
Find the sums given below:
(i) − 5 + (− 8) + (− 11) + ………… + (− 230) Solution: Given to us that, (−5) + (−8) + (−11) + ………… + (−230) For the above given A.P., The first term, a = −5 The nth term, an= −230 Common...
Find the sums given below:
(i) $7+10 \frac{1}{2}+14+\ldots \ldots \ldots \ldots \ldots . .+84$ (ii) $34+32+30+\ldots \ldots \ldots . .+10$ Solutions: (i) For the given A.P., $7+10 \frac{1}{2}+14+\ldots \ldots \ldots \ldots...
A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of 1 4 m and a tread of 1 2 m. (see Fig. 5.8)Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step = ¼ ×1/2 ×50 m3.]
Solution: The first step is $\frac{1}{2}$ m wide, the second step is 1 m wide, and the third step is $\frac{3}{2}$m wide, as seen in the diagram. When the height reaches $\frac{1}{4}$ m, we can see...
The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. [Hint :Sx – 1 = S49 – Sx ]
Solution: Given, Row houses have numbers ranging from 1,2,3,4,5,......49. As a result, we can observe that the houses in a row are in the form of AP. So, The first term, a = 1 The common difference,...
A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are apart, what is the length of the wood required for the rungs? [Hint: Number of rungs = -250/25 ].
Solution: Given, The ladder has a 25cm distance between the rungs. The distance between the ladder's top and bottom rungs of ladder is $=2\frac{1}{2}m=2\frac{1}{2}*100cm$= 5/2 ×100cm = 250cm As a...
The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.
Solution: We can write from the given statements, a3 + a7 = 6 …………………………….(i) And a3 ×a7 = 8 ……………………………..(ii) By using the nth term formula, an = a+(n−1)d Third...
Which term of the AP: 121, 117, 113, . . ., is its first negative term? [Hint: Find n for an < 0]
Solution: Given that the AP series is 121, 117, 113, . . ., The, first term, a = 121 and, The common difference, d = 117-121= -4 According to the nth term formula, an = a+(n −1)d...
Find the sum of the following APS.
(i) $0.6,1.7,2.8, \ldots \ldots \ldots$, to 100 terms
(ii) $1 / 15,1 / 12,1 / 10, \ldots \ldots$, to 11 terms
Solutions: (i) Provided that, $0.6,1.7,2.8, \ldots$, to 100 terms And for this A.P., The first term, $a=0.6$ And the common difference, $d=a_{2}-a_{1}=1.7-0.6=1.1$ We all know that,for the sum of...
Find the sum of the following APS.
(i) $2,7,12, \ldots .$, to 10 terms. (ii) $-37,-33,-29, \ldots$, to 12 terms Solutions: (i) Provided that, $2,7,12, \ldots$, to 10 terms And for this A.P., The first term, $\mathrm{a}=2$ And the...
Ramkali saved Rs 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in the nth week, her weekly savings become Rs 20.75, find n.
Given this, Ramkali saved Rs.5 in the first week and then began saving Rs.1.75 each week after that. As a result, the first term is a = 5 and the common difference is d = 1.75. In addition,...
Subba Rao started work in 1995 at an annual salary of Rs 5000 and received an increment of Rs 200 each year. In which year did his income reach Rs 7000?
Solution: As can be observed from the provided question, Subba Rao's earnings increase by Rs.200 every year, forming an AP. As a result, after 1995, the annual salaries are as follows: 5000, 5200,...
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the A.P.
Solution: We all know what the AP's nth phrase is; an = a+(n−1)d a4 = a+(4−1)d a4 = a+3d We can write in the same way., a8 = a+7d a6 = a+5d...
Find the 20th term from the last term of the A.P. 3, 8, 13, …, 253.
Solution: If A.P. is 3, 8, 13,..., 253, then the common difference is d= 5. As a result, we can rewrite the given AP as follows: 253, 248, 243, …, 13, 8, 5 For the new AP, the first term is a = 253...
Determine the A.P. whose third term is 16 and the 7th term exceeds the 5th term by 12.
Solutions: Provided here that, The Third term, a3 = 16 As we all know, a +(3−1)d = 16 a+2d = 16 ………………………………………. (i) It is known that, 7th term exceeds the...
For what value of n, are the nth terms of two APs 63, 65, 67, and 3, 10, 17, … equal?
Solution: Two APs are given: 63, 65, 67,... and 3, 10, 17,.... Taking first AP, 63, 65, 67, … The First term, a = 63 The Common difference, d = a2−a1 = 65−63 = 2 We all know that...
How many multiples of 4 lie between 10 and 250?
Solution: The number 12 is the first multiple of four that is greater than ten. The following multiple will be 16. As a result, the series formed as follows: 12, 16, 20, 24,... All of these are...
How many three digit numbers are divisible by 7?
Solution: The first three-digit number that is divisible by seven is 105. 105+7 = 112 is the second number. 112+7 = 119 is the third number. As a result, 105, 112, 119,... All three digit integers...
Two APs have the same common difference. The difference between their 100th term is 100, what is the difference between their 1000th terms?
Solution: Let a1 and a2 be the initial terms of two APs, respectively. And let d be the common difference between these APs. For the first A.P.,we know, an = a+(n−1)d Therefore,...
Which term of the A.P. 3, 15, 27, 39,.. will be 132 more than its 54th term?
Solution: Given to us that A.P. is 3, 15, 27, 39, … The first term, a = 3 The common difference, d = a2 − a1 = 15 − 3 = 12 We all know that, an = a+(n−1)d...
If 17th term of an A.P. exceeds its 10th term by 7. Find the common difference.
Solution: For an A.P series we all know that; an = a+(n−1)d a17 = a+(17−1)d a17 = a +16d Similarly, a10 = a+9d As stated in the question,...
If the 3rd and the 9th terms of an A.P. are 4 and − 8 respectively. Which term of this A.P. is zero?
Solution: Provided that, a3 = 4 is the 3rd term and, a9 = −8 is the 9th term We all know that, an = a+(n−1)d Therefore, a3 = a+(3−1)d 4...
An A.P. consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
Solution: Provided that, a3 = 12 is the 3rd term a50 = 106 is the 50th term We all know that, ${{a}_{n}}=a+(n-1)d$ ${{a}_{3}}=a+(3-1)d$ 12...
Find the 31st term of an A.P. whose 11th term is 38 and the 16th term is 73.
Solution: Provided that, For the 11th term, a11 = 38 and for the16th term, a16 = 73 We all know that, ${{a}_{n}}=a+(n-1)d$ a11 = a+(11−1)d 38...
Check whether -150 is a term of the A.P. 11, 8, 5, 2, …
Solution: In the above mentioned given series, A.P. 11, 8, 5, 2.. The first term, a = 11 The common difference, d = a2−a1 = 8−11 = −3 Let −150 be the nth term of this...
Find the number of terms in each of the following A.P.
(i) 7, 13, 19, …, 205 (ii)$18,15\frac{1}{2},13...-47$ Solutions: (i) Given here, 7, 13, 19, …, 205 is the A.P Therefore The first term, a = 7 The common difference,...
Which term of the A.P. 3, 8, 13, 18, … is 78?
Solutions: The A.P. series is given as 3, 8, 13, 18, … The first term, a = 3 The common difference, d = a2 − a1 = 8 − 3 = 5 Let the nth term of the given...
In the following APs find the missing term in the boxes.
Solution: (i) In the case of given A.P., a2 = 38 a6 = −22 As we all know, for an A.P., an = a+(n −1)d Therefore, putting the values here, a2 = a+(2−1)d 38...
In the following APs find the missing term in the boxes.
Solutions: (i) In the case of given A.P., a = 5 and a4 = 19/2 As we all know, for an A.P., an = a+(n−1)d Therefore, putting the values here, a4 = a+(4-1)d 19/2...
In the following APs find the missing term in the boxes.
Solutions: (i) In the case of the given A.P., 2,2 , 26 First and the third term are; a = 2 a3 = 26 As we all know, for an A.P., an = a+(n −1)d Therefore, putting the...
Choose the correct choice in the following and justify:
(i) 30th term of the A.P: 10,7, 4, …, is(A) 97 (B) 77 (C) −77 (D) −87 (ii) 11th term of the A.P. -3, -1/2, ,2 …. is(A) 28 (B) 22 (C) – 38 (D)$-48\frac{1}{2}$ Solutions: (i)...
Fill in the blanks in the following table, given that a is the first term, d the common difference and an the nth term of the A.P.
(i) Given here, The first term, a = 3.5 The common difference, d = 0 Number of terms given, n = 105 Nth term, an = ? As we all know, for an A.P.,...
Fill in the blanks in the following table, given that a is the first term, d the common difference and an the nth term of the A.P.
(i) Given here, The first term, a = ? The common difference, d = -3 Number of terms given, n = 18 Nth term, an = -5 As we all know, for an A.P.,...
Fill in the blanks in the following table, given that a is the first term, d the common difference and an the nth term of the A.P.
Solutions: (i) Given here, The first term, a = 7 The common difference, d = 3 Number of terms given, n = 8 We need to seek out the nth term, ${{a}_{n}}=?$ As we all know, for an A.P.,...
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(i) 12, 52, 72, 73 … Solution(i): Given here, ${{a}_{2}}-{{a}_{1}}=25-1=24$ ${{a}_{3}}-{{a}_{2}}=49-25=24$ ${{a}_{4}}-{{a}_{3}}=73-49=24$ Since, an+1 – an or the common...
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(i) √3, √6, √9, √12 … (ii) 12, 32, 52, 72 … Solution(i): Given here, ${{a}_{2}}-{{a}_{1}}=\sqrt{6}-\sqrt{3}=\sqrt{3}*\sqrt{2}-\sqrt{3}=\sqrt{3}(\sqrt{2}-1)$...
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(i) a, a2, a3, a4 … (ii) √2, √8, √18, √32 … Solution(i): Given here, ${{a}_{2}}-{{a}_{1}}={{a}^{2}}-a=a(a-1)$ ${{a}_{3}}-{{a}_{2}}={{a}^{3}}-{{a}^{2}}={{a}^{2}}(a-1)$...
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(i) 1, 3, 9, 27 … (ii) a, 2a, 3a, 4a … Solution(i):Given here, ${{a}_{2}}-{{a}_{1}}=3-1=2$ ${{a}_{3}}-{{a}_{12}}=9-3=6$ ${{a}_{4}}-{{a}_{3}}=27-9=18$ Since, an+1 – an or the...
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(i) 0, -4, -8, -12 … (ii) -1/2, -1/2, -1/2, -1/2 …. Solution(i):Given here, ${{a}_{2}}-a=(-4)-0=-{{4}_{{}}}$ ${{a}_{3}}-{{a}_{2}}=(-8)-(-4)=-{{4}_{{}}}$...
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(i) Given, 3, 3+√2, 3+2√2, 3+3√2(ii) 0.2, 0.22, 0.222, 0.2222 …. Solution(i):Given here, ${{a}_{2}}-a=3=\sqrt{2}-3={{\sqrt{2}}_{{}}}$...
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(i) Given, -1.2, – 3.2, -5.2, -7.2 … (ii) Given, -10, – 6, – 2, 2 … Solution(i): Given here, ${{a}_{2}}-a=(-3.2)-(1.2)=-{{2}_{{}}}$ ${{a}_{3}}={{a}_{2}}=(-5.2)-(-3.2)=-2$...
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(i) 2, 4, 8, 16 … (ii) 2, 5/2, 3, 7/2 …. Solution(i): It was given to us that, 2, 4, 8, 16 … The main difference here is; ${{a}_{2}}-{{a}_{1}}=4-2=2$ ${{a}_{3}}-{{a}_{2}}=8-4=4$...
For the following A.P.s, write the first term and the common difference.
(i) 1/3, 5/3, 9/3, 13/3 …. (ii) 0.6, 1.7, 2.8, 3.9 … Solution(i): The first term is, a = 1/3 And, the Common difference, d = Second term – First term Therefore, ⇒ 5/3 – 1/3...
For the following A.P.s, write the first term and the common difference.
(i) 3, 1, – 1, – 3 … (ii) -5, – 1, 3, 7 … Solution(i): 3, 1, – 1, – 3 … The first term is, a = 3 And, the Common difference, d = Second term – First term Therefore, ⇒ 1 – 3...
Write first four terms of the A.P. when the first term a and the common difference is given as follows:
(i) a = – 1.25, d = – 0.25 Solution: Assume that the Arithmetic Progression series is a1, a2, a3, a4, a5 … a1 = a = – 1.25...
Write first four terms of the A.P. when the first term a and the common difference are given as follows:
(i) a = 4, d = – 3 (ii) a = -1 d = 1/2 Solution(i): Assume that the Arithmetic Progression series is a1, a2, a3, a4, a5 …...
Write first four terms of the A.P. when the first term a and the common difference are given as follows:
(i) a = 10, d = 10 (ii) a = -2, d = 0 Solution(i): Assume that the Arithmetic Progression series is a1, a2, a3, a4, a5 … a1 = a =...
In which of the following situations, does the list of numbers involved make as arithmetic progression and why?
(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km. (ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air...
In which of the following situations, does the list of numbers involved make as arithmetic progression and why?
(i) The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre. (ii) The amount of money in the account every...
In which of the following situations, does the list of numbers involved make as arithmetic progression and why?
(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km. (ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air...