Answer:
See below.
Step-by-step 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.
