Question

Use mathematical induction to prove that the statement is true for every positive integer n. 8 + 16 + 24 + . . . + 8n = 4n(n + 1)

Answer 1

First show the statement holds for $n=1$. The left hand side is just 8, and the right hand side is $4(1)(1+1)=8$, so it's true.

Assume the statement holds for $n=k$, i.e.

$8+16+\cdots+8(k-1)+8k=4k(k+1)$

and use this to show it holds for $n=k+1$, i.e.

$8+16+\cdots+8(k-1)+8k+8(k+1)=4(k+1)(k+2)$

By the assumption above, you have

$\underbrace{8+16+\cdots+8(k-1)+8k}_{n=k}+8(k+1)=4k(k+1)+8(k+1)=4(k+1)(k+2)$

so the statement is true for all $n\ge1$.