Question

We mathematical induction to prove the statement is true for all positive integers n. 8+16+24+...+8n=4n(n+1)

Answer 1

Assume its true for n = k 
then 8 + 16 + 24+...  + 8k = 4k(k + 1)

sum for k+1 terms = 4k(k + 1) + 8(k + 1)
   =   4(k + 1)( (k + 2)
   =  4(k + 1)( k + 1 + 1)

Now this is the same as the formula for k terms with the k being replaced by k + 1 . So if the formula is correct for k terms then its must also be correct for k+1 terms
Also its true for k = 1 because 8 = 4(1)(1 + 1)  so its true for k = 2,3 4 and so
on.  - for all values of k ( n).

This compleres the proof.