Question -
Answer -
Let P (n): 32n + 2 – 8n – 9 is divisible by8
Let us check for n = 1,
P (1): 32.1 + 2 – 8.1 – 9
: 81 – 17
: 64
P (n) is true for n = 1. Where, P (n) is divisible by 8
Now, let us check for P (n) is true for n = k, and have toprove that P (k + 1) is true.
P (k): 32k + 2 – 8k – 9 is divisible by 8
: 32k + 2 – 8k – 9 = 8λ … (i)
We have to prove,
32k + 4 – 8(k + 1) – 9 is divisible by 8
3(2k + 2) + 2 – 8(k + 1) – 9 = 8μ
So,
= 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
Hence, P (n) is true for all n ∈ N.