Prove the following by the principle of mathematical induction:
32n + 7 is divisible by 8 for all n ϵ N
Let P(n): 32n + 7 is divisible by 8
Let’s check For n =1
P(1): 32 + 7 = 9 + 7
= 16
Since, it is divisible by 8
So, P(n) is true for n=1
Now, for n=k
P(k): 32k + 7 = 8λ - - - - - - - (1)
We have to show that,
32(k + 1) + 7 is divisible by 8
32k + 2 + 7 = 8μ
Now,
32(k + 1) + 7
= 32k.32 + 7
= 9.32k + 7
= 9.(8λ - 7) + 7
= 72λ - 63 + 7
= 72λ - 56
= 8(9λ - 7)
= 8μ
Therefore, P(n) is true for n = k + 1
Hence, P(n) is true for all n∈N by PMI