20. <math><mrow><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo></mrow></math> 1 is divisible by 11.

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Ncert Solutions Maths class 11th Maths Class 11th Principle of Mathematical Induction

20. $1 0^{2 n - 1} +$ 1 is divisible by 11.

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Payal Gupta | Contributor-Level 10

4 months ago

20. LetP(n): $1 0^{2 n - 1} +$ 1 is divisible by 11.

Putting n = 1

$P (1) = 1 0^{'} + 1 = 11$ is divisible by 11.

Which is true. Thus, P(1) is true.

Let us assume that P(k) is true for some natural no. k.

P(k)= $1 0^{2 k - 1} +$

$1 is divisible by 11.$

$1 0^{2 k - 1} + 1 = 11 a a \in z$

$\to 1 0^{2 k - 1} = 11 a - 1$ (1)

we want to prove that P(k +1) is true.

$P (k + 1) : 1 0^{2 (K + 1) - 1} + 1 = 1 0^{2 k + 1} + 1 is divisible by 11.$

$1 0^{2 k + 1} + 1 = 1 0^{(2 k - 1) + 2} + 1$

$= 1 0^{2 k - 1} \cdot 1 0^{2} + 1$

$= 100 (11 a - 1) + 1 (using(1))$

=1100a $-$ 99= 11(100a $-$ 9)

11b where b= (100a $-$ 9) $\in z$

$\Rightarrow 1 {0^{2}}^{k + 1} + 1$ is divisible by 11.

$\Rightarrow P (k + 1)$

is true when p(k) is true.

Hence by P.M.I. P(n) is true for every positive integer.

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