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