Question

Use mathematical induction to prove that the statement is true for every positive integer n.

Answer 1

Induction on n

Base case: n = 0

the left side is 0, and the right side is 0, so the equality holds.

Now, suppose that is true for n, and you need to show that the equality holds for n+1

So, write the left side

8 + 16 +24 + ... + 8n + 8(n+1) (*)

By hypothesis, 8+16+24+ ...+ 8n = 4n(n+1)

so, in (*) we have

4n(n+1) +8(n+1) = (4n+8)(n+1) = 4(n+2)(n+1) = 4(n+1)(n+2). So, this is true for every natural number n.