Answer: Given, 24th term is twice the 10th term a24 = 2a10 an = a + (n – 1) d When n = 10, a10 = a + (10 – 1)d = a + 9d When n = 24, a24 = a + (24 – 1)d = a + 23d When n = 34, a34 = a + (34 – 1)d =...
The 10th and 18th term of an A.P. are 41 and 73 respectively, find 26th term.
Answer: Given, 10th term of an A.P is 41, and 18th terms of an A.P. is 73 a10 = 41 a18 = 73 an = a + (n – 1) d When n = 10, a10 = a + (10 – 1)d = a + 9d When n = 18, a18 = a + (18 – 1)d = a + 17d...
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that the 25th term of the A.P. is Zero.
Answer: Given, 10 times the 10th term of an A.P. is equal to 15 times the 15th term 10a10 = 15a15 an = a + (n – 1) d When n = 10, a10 = a + (10 – 1)d = a + 9d When n = 15, a15 = a + (15 – 1)d = a +...
If 9th term of an A.P. is Zero, prove that its 29th term is double the 19th term.
Answer: Given, 9th term of an A.P is 0 a9 = 0 an = a + (n – 1) d [w When n = 9, a9 = a + (9 – 1)d = a + 8d a9 = 0 a + 8d = 0 a = -8d When n = 19, a19 = a + (19 – 1)d = a + 18d = -8d + 18d =...
The 6th and 17th terms of an A.P. are 19 and 41 respectively. Find the 40th term.
Answer: Given, 6th term of an A.P is 19 and 17th terms of an A.P. is 41 a6 = 19 a17 = 41 an = a + (n – 1) d When n = 6, a6 = a + (6 – 1) d = a + 5d When n = 17, a17 = a + (17 – 1)d = a + 16d a6 = 19...
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.
Answer: Given, First term, a = 5; last term, l = an = 80 Common difference, d = 3 an = a + (n – 1) d an = 5 + (n – 1)3 = 5 + 3n – 3 = 3n + 2 Put an = 80 as 80 is last term of A.P. 3n + 2 = 80 3n =...
How many terms are there in the A.P. -1, -5/6, -2/3, -1/2, …, 10/3 ?
Answer: Given, AP of -1, -5/6, -2/3, -1/2, … a1 = a = -1 a2 = -5/6 Common difference, d = a2 – a1 = -5/6 – (-1) = -5/6 + 1 = (-5+6)/6 = 1/6 an = a + (n – 1) d an = -1 + (n – 1) 1/6 = -1 + 1/6n – 1/6...
How many terms are in A.P. 7, 10, 13,…43?
Answer: Given, AP of 7, 10, 13,… a1 = a = 7 a2 = 10 Common difference, d = a2 – a1 = 10 – 7 = 3 an = a + (n – 1) d an = 7 + (n – 1)3 = 7 + 3n – 3 = 3n + 4 Put, an = 43 3n + 4 = 43 3n = 43 – 4 3n =...
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, … is (a) purely real (b) purely imaginary ?
Answer: Given, AP of 12 + 8i, 11 + 6i, 10 + 4i, … a1 = a = 12 + 8i a2 = 11 + 6i Common difference, d = a2 – a1 = 11 + 6i – (12 + 8i) = 11 – 12 + 6i – 8i = -1 – 2i an = a + (n – 1) d an = 12 + 8i +...
Which term of the sequence 24, 23 ¼, 22 ½, 21 ¾ is the first negative term?
Answer: Given, AP of 24, 23 ¼, 22 ½, 21 ¾, … = 24, 93/4, 45/2, 87/4, … a1 = a = 24 a2 = 93/4 Common difference, d = a2 – a1 = 93/4 – 24 = (93 – 96)/4 = – 3/4 an = a + (n – 1) d an = a + (n – 1) d...
(i) Is 68 a term of the A.P. 7, 10, 13,…? (ii) Is 302 a term of the A.P. 3, 8, 13,…?
Answers: (i) Given, A.P is 7, 10, 13,… a1 = a = 7 a2 = 10 Common difference, d = a2 – a1 = 10 – 7 = 3 an = a + (n – 1)d an = 7 + (n – 1)3 = 7 + 3n – 3 = 3n + 4 Put, an = 68 3n + 4 = 68 3n = 68 – 4...
Which term of the A.P. 4, 9, 14,… is 254 ?
Answer: Given, A.P is 4, 9, 14,… a1 = a = 4 a2 = 9 Common difference, d = a2 – a1 = 9 – 4 = 5 an = a + (n – 1)d an = 4 + (n – 1)5 = 4 + 5n – 5 = 5n – 1 Put, an = 254 5n – 1 = 254 5n = 254 + 1 5n =...
(i) Which term of the A.P. 3, 8, 13,… is 248 ? (ii) Which term of the A.P. 84, 80, 76,… is 0 ?
Answer: (i) Given, A.P is 3, 8, 13,… a1 = a = 3 a2 = 8 Common difference, d = a2 – a1 = 8 – 3 = 5 an = a + (n – 1)d an = 3 + (n – 1)5 = 3 + 5n – 5 = 5n – 2 (Put, an = 248) ∴ 5n – 2 = 248 = 248 + 2...
In an A.P., show that am+n + am–n = 2am.
Answer: Using the formula, an = a + (n – 1)d LHS: am+n + am-n am+n + am-n = a + (m + n – 1)d + a + (m – n – 1)d = a + md + nd – d + a + md – nd – d = 2a + 2md – 2d = 2(a + md – d) = 2[a + d(m – 1)]...
Find: nth term of the A.P 13, 8, 3, -2, ….
Answer: nth term of the A.P 13, 8, 3, -2, …. Arithmetic Progression (AP) whose common difference is = an – an-1 where n > 0 Consider, a = a1 = 13, a2 = 8 … Common difference, d = a2 – a1 = 8 – 13...
Find: (i) 10th term of the A.P. 1, 4, 7, 10, ….. (ii) 18th term of the A.P. √2, 3√2, 5√2, …
Answer: (i) Arithmetic Progression (AP) whose common difference is = an – an-1 where n > 0 Consider, a = a1 = 1, a2 = 4 … Common difference, d = a2 – a1 = 4 – 1 = 3 Finding an an = a + (n-1) d =...