Let P(n):\[1\text{ }+\text{ }2\text{ }+\text{ }{{2}^{2}}~+\text{ }\ldots \text{ }+\text{ }{{2}^{n}}~=\text{ }{{2}^{n}}{{~}^{+1}}~\text{ }1\] , for all natural numbers n \[P\left( 1 \right):\text{...
Prove 2 + 4 + 6 + …+ 2n = n^2 + n for all natural numbers n.
As indicated by the inquiry, \[P\left( n \right)\text{ }is\text{ }2\text{ }+\text{ }4\text{ }+\text{ }6\text{ }+\text{ }\ldots \text{ }+\text{ }2n\text{ }=\text{ }n^2\text{ }+\text{...
Prove √n < 1/√1 + 1/√2 + … 1/√n, for all natural numbers n ≥ 2.
As per the inquiry, \[\Rightarrow P\left( k+1 \right)\] is valid when P(k) is valid. Along these lines, by Mathematical Induction, \[\surd n\text{ }<\text{ }1/\surd 1\text{ }+\text{ }1/\surd...
Prove 2n < (n + 2)! for all natural number n.
As indicated by the inquiry, \[P\left( n \right)\text{ }is\text{ }2n\text{ }<\text{ }\left( n\text{ }+\text{ }2 \right)!\] In this way, subbing various qualities for n, we get, \[P\left( 0...
Prove n^2 < 2n for all natural numbers n ≥ 5.
As indicated by the inquiry, \[P\left( n \right)\text{ }is\text{ }n\hat{\ }^2\text{ }<\text{ }2n\text{ }for\text{ }n\ge 5\] Let \[P\left( k \right)\text{ }=\text{ }k^2\text{ }<\text{ }2k\] be...
Prove n^3 – n is divisible by 6, for each natural number n ≥ 2.
As indicated by the inquiry, \[P\left( n \right)\text{ }=\text{ }n^3\text{ }-\text{ }n\] is distinct by 6. Along these lines, subbing various qualities for n, we get, \[P\left( 0 \right)\text{...
Prove For any natural number n, 7^n – 2^n is divisible by 5.
As indicated by the inquiry, \[~P\left( n \right)\text{ }=\text{ }7n\text{ }\text{ }2n\] is distinct by 5. In this way, subbing various qualities for n, we get, \[~P\left( 0 \right)\text{ }=\text{...
Prove 3^2n – 1 is divisible by 8, for all natural numbers n.
As indicated by the inquiry, \[P\left( n \right)\text{ }=\text{ }32n\text{ }\text{ }1\] is distinct by 8. Along these lines, subbing various qualities for n, we get, \[P\left( 0 \right)\text{...
Prove n^3 – 7n + 3 is divisible by 3, for all natural numbers n.
As per the inquiry, \[P\left( n \right)\text{ }=\text{ }n3\text{ }\text{ }7n\text{ }+\text{ }3\] is distinguishable by 3. Along these lines, subbing various qualities for n, we get, \[P\left( 0...
Prove 2^3n – 1 is divisible by7, for all natural numbers n.
As indicated by the inquiry, \[P\left( n \right)\text{ }=\text{ }23n\text{ }\text{ }1\] is detachable by 7. In this way, subbing various qualities for n, we get, \[P\left( 0 \right)\text{ }=\text{...
Give an example of a statement P(n) which is true for all n. Justify your answer.
As indicated by the inquiry, P(n) which is valid for all n. Let P(n) be, ⇒ P(k) is valid for all k. In this manner, P(n) is valid for all n.
Give an example of a statement P(n) which is true for all n ≥ 4 but P(1), P(2) and P(3) are not true. Justify your answer.
As indicated by the inquiry, P(n) which is valid for all n ≥ 4 however P(1), P(2) and P(3) are false Let \[P\left( n \right)\text{ }be\text{ }2n\text{ }<\text{ }n!\] Thus, the instances of the...
Prove the following by using the principle of mathematical induction for all n ∈ N: (2n +7) < (n + 3)2
We can compose the given assertion as \[P\left( n \right):\text{ }\left( 2n\text{ }+7 \right)\text{ }<\text{ }\left( n\text{ }+\text{ }3 \right)2\] In the event that \[n\text{ }=\text{ }1\]we get...
Prove the following by using the principle of mathematical induction for all n ∈ N: 41n – 14n is a multiple of 27
We can compose the given assertion as $P(n):{{41}^{n}}-{{14}^{n}}$is a various of \[27\] In the event that \[n\text{ }=\text{ }1\]we get $P(1)={{41}^{1}}-{{14}^{1}}=27$, which is a various by...
Prove the following by using the principle of mathematical induction for all n ∈ N: 32n + 2 – 8n – 9 is divisible by 8
We can compose the given assertion as $P(n):{{3}^{2}}n+2-8n-9$is separable by \[8\] In the event that \[n\text{ }=\text{ }1\]we get $P(1)={{3}^{(2\text{ }\!\!\times\!\!\text{ }1)}}+2-8\text{ ...
Prove the following by using the principle of mathematical induction for all n ∈ N: x^2n – y^2n is divisible by x + y
. We can compose the given assertion as $P(n):{{x}^{2}}n-{{y}^{2}}n$is distinguishable by \[x\text{ }+\text{ }y\] In the event that \[n\text{ }=\text{ }1\]we get $P(1)={{x}^{2}}\text{ ...
Prove the following by using the principle of mathematical induction for all n ∈ N: 10^(2n – 1) + 1 is divisible by 11
We can compose the given assertion as $P(n):{{10}^{2}}n-1+1$is distinct by \[11\] On the off chance that \[n\text{ }=\text{ }1\]we get $P(1)={{10}^{2}}.1-1+1=11$, which is distinct by \[11\] Which...
Prove the following by using the principle of mathematical induction for all n ∈ N: n (n + 1) (n + 5) is a multiple of 3
We can compose the given assertion as \[P\text{ }\left( n \right):\text{ }\left( n\text{ }+\text{ }1 \right)\text{ }\left( n\text{ }+\text{ }5 \right)\], which is a different of \[3\] In the...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- We can compose the given assertion as \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- By additional improvement \[=\text{ }\left( \mathbf{k}\text{ }+\text{ }\mathbf{1} \right)\text{ }+\text{ }\mathbf{1}\] \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- We can compose the given assertion as \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical...
Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2
We can compose the given assertion as \[\mathbf{P}\text{ }\left( \mathbf{n} \right):\text{ }\mathbf{1}.\mathbf{2}\text{ }+\text{ }\mathbf{2}.\mathbf{22}\text{ }+\text{ }\mathbf{3}.\mathbf{22}\text{...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
Solution: \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left( n...