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

to prove 8 + 16 + 24 + ... + 8 n = 4 n (n + 1) - - - - (*) let T n = 4 n (n + 1) (1) verify for n = 1 L H S = 8 R H S = 4 * 1 (1 + 1) = 4 * 2 = 8 ∴ true for n = 1 # to show T k ⇒ T k + 1 assume true for T k = 4 k (k + 1) need to show T k + 1 = 4 (k + 1) (k + 2) add next term to to both sides of (*) 8 + 16 + 24 + ... + 8 k + 8 (k + 1) = 4 k (k + 1) + 8 (k + 1) ∴ T k + 1 = 4 k (k + 1) + 8 (k + 1) = 4 (k + 1) [ k + 2 ] = T k + 1 i. e. T k ⇒ T k + 1 as required # (3) conclusion statement true for T 1 ∵ T k ⇒ T k + 1 T 1 ⇒ T 2 ⇒ T 3 ⇒ ... ∀ n ∈ N