Question

Prove by induction that 7^2n+1 +1 is divisible by 8, for all nEN

Answer 1

See below.

Explanation:

Base case:

Replace n with 1.

7^(2×1+1)+1

7^3+1

343+1

344

8 is a factor of 344 since 344=8(43).

Induction hypothesis:

Assume there is some integer n such that 7^(2k+1)+1=8n for positive integer k.

7^(2[k+1]+1)+1

7^(2k+3)+1

7^(2k+1+2)+1

7^(2k+1)7^2+1

49×7^(2k+1)+1

Induction step:

49×(8n-1)+1

49(8n)-49+1

49(8n)-48

8[49n-6]

This means 8 is a factor of 7^(2(k+1)+1)+1.

Thus, this proves for all positive integer n that 8 is a factor of 7^(2n+1)+1.