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in Mathematical Induction by (50.8k points)

Prove by the principle of mathematical induction:

 32n + 2 – 8n – 9 is divisible by 8 for all n ϵ N

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Suppose P (n): 32n + 2 – 8n – 9 is divisible by 8

Now let us check for n = 1,

P (1): 32.1 + 2 – 8.1 – 9

: 81 – 17

: 64

P (n) is true for the n = 1. Where, P (n) is divisible by 8

Then, let us check for P (n) is true for n = k, and have to prove that P (k + 1) is true.

P (k): 32k + 2 – 8k – 9 is divisible by 8

: 32k + 2 – 8k – 9 = 8λ … (i)

Now we have to prove,

32k + 4 – 8(k + 1) – 9 is divisible by 8

3(2k + 2) + 2 – 8(k + 1) – 9 = 8μ

Therefore,

= 32(k + 1).32 – 8(k + 1) – 9

= (8λ + 8k + 9)9 – 8k – 8 – 9

= 72λ + 72k + 81 – 8k – 17 using equation (1)

= 72λ + 64k + 64

= 8(9λ + 8k + 8)

= 8μ

P (n) is true for n = k + 1

Thus, P (n) is true for all n ∈ N.

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