Question -
Answer -
Let P (n): 32n + 7 is divisible by 8
Let us check for n = 1,
P (1): 32 + 7 = 9 + 7 = 16
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 + 7 is divisible by 8
: 32k + 7 = 8λ
: 32k = 8λ – 7 … (i)
We have to prove,
32(k + 1) + 7 is divisible by 8
32k + 2 + 7 = 8μ
So,
= 32(k + 1) + 7
= 32k.32 + 7
= 9.32k + 7
= 9.(8λ – 7) + 7 by using equation (i)
= 72λ – 63 + 7
= 72λ – 56
= 8(9λ – 7)
= 8μ
P (n) is true for n = k + 1
Hence, P (n) is true for all n ∈ N.