22. <math><mrow><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>8</mn><mi>n</mi><mo>−</mo><mn>9</mn></mrow></math> is divisible by 8

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

22. $3^{2 n + 2} - 8 n - 9$ is divisible by 8

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Payal Gupta | Contributor-Level 10

4 months ago

22. Let P(n): $3^{2 n + 2} - 8 n - 9$ is divisible by 8

put n= 1,

P(1): $3^{2 + 2} - 8.1 - 9$

3⁴ – 8 – 9 = 81– 17 = 64= is divisible by 8

Which is true.

Assume that P(k) is true for some natural numbers k.

i.e, $3^{2 k + 2} - 8 k - 9$ be divisible by 8

$3^{2 k + 2} - 8 k - 9 = 8 a$ where,a $\in z$

$3^{2 k + 2} = 8 a + 8 k + 9$ (1)

We want to prove thatP(k+ 1) is true.

$P (k + 1) : 3^{2 (k + 1) + 2} - 8 (k + 1) - 9$ is divisible by 8, is also true.

Now,

${3^{2 k + 2 +}}^{2} - = 8 (k + 1) - 9$

= $3^{2 k + 4} - 8 k + 8 - 9$

$3^{(2 k + 2) + 2} - 8 k + 8 - 9$

3^{(2k +2)}. 3² $-$ 8k $-$ 17

$= 9 (8 a + 8 k + 9) - 8 k - 17$ (Using 1)

$= 72 a + 72 k + 81 - 8 k - 17$

= 72a + 64k+ 64 = 8(9a + 8k + 8)

= 8b, where b = 9a + 8b + 8 $a \in z$

$\Rightarrow$ 3^{2k + 4}– 8(k+1) – 9 is divisible by 8.

$∴$ P(k+1) is true when P(k) is true. Hence, By P.M.I. P(n) is true for all positive integer n.

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24. Let P (n) be the statement “ 2n+7< (n+3)2”

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$\Rightarrow P (k + 1)$ is true.

$∴$ By the principle of mathematical induction, P (n)is true for all n $\in$ N.

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