P n - a 2 n - b 2 n is divisible by a - b - - n - NTo prove P n using mathematical induction- the base case isA- a1-b1 is divisible by a-bB- a2-b2-a-ba-b is divisible by a-bC- a2 k-b2 k is divisible by a-b D- a k - b k is divisible by a - b

The correct option is B a^2 b^2=(a+b)(a b) is divisible by a+b P(n):a^2n b^2n is divisible by a+b, ∀ n ϵ N Since we need to prove P(n) for all natural number ...

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Standard XI

Mathematics

Proof by mathematical induction

P n : a 2 n -...

Question

$P (n) : a^{2 n} - b^{2 n} is divisible by a + b, \forall n ϵ N$

To prove P(n) using mathematical induction, the base case is

$a^{2} - b^{2} = (a + b) (a - b) is divisible by a + b$

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$a^{2 k} - b^{2 k} is divisible by a + b$

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$a^{k} - b^{k} is divisible by a + b$

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$a^{1} - b^{1} is divisible by a + b$

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Solution

The correct option is A
$a^{2} - b^{2} = (a + b) (a - b) is divisible by a + b$

$P (n) : a^{2 n} - b^{2 n} is divisible by a + b, \forall n ϵ N$

Since we need to prove P(n) for all natural numbers, the base case will be n=1.
Substituting n=1, we get
$a^{2} - b^{2} = (a + b) (a - b) is divisible by a + b$

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Similar questions

By principle of mathematical Induction show that aⁿ-bⁿ is divisible by a+b when n is a positive integer.

Q. A four digit number abcd is divisible by 11 if only
A.(a+b)-(c+d) is divisible by 11
B.(a-c)-(b-d) is divisible by 11
C.(a-c)-(b+d) is divisible by 11
D.(b+d)-(a+c) is divisible by 11
The coerrect answer is 'D' but how?

a \equiv b (m o d n)

means .

Q. By mathematical induction

p^{n + 1} + (p + 1)^{2 n - 1}

is divisible by

Q. Assertion :(A):

P (n) = 7^{n} - 3^{n}

is divisible by

4

. Reason: (R): If

a \neq b,

then expansion

a^{n} - b^{n}

is divisible by

a - b

n

is odd or even.

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P(n):a2n−b2n is divisible by a+b, ∀ n ϵ N To prove P(n) using mathematical induction, the base case is

The correct option is A a2−b2=(a+b)(a−b) is divisible by a+b P(n):a2n−b2n is divisible by a+b, ∀ n ϵ N Since we need to prove P(n) for all natural numbers, the base case will be n=1. Substituting n=1, we get a2−b2=(a+b)(a−b) is divisible by a+b

$P (n) : a^{2 n} - b^{2 n} is divisible by a + b, \forall n ϵ N$

To prove P(n) using mathematical induction, the base case is