Proof-by-mathematical-induction-that-f-n-n-3-5n-is-a-multiple-of-6-

Question Number 137875 by physicstutes last updated on 07/Apr/21

Proof by mathematical induction that

f (n) = n^{3} + 5 n

is a multiple of 6 .

Answered by mathmax by abdo last updated on 08/Apr/21

f (0) = 0 is mulyiple of 6 let suppose f (n) multiple of 6 \Rightarrow

f (n) = 6 k \Rightarrow n^{3} + 5 n = 6 k

f (n + 1) = {(n + 1)}^{3} + 5 (n + 1) = n^{3} + {3 n}^{2} + 3 n + 1 + 5 n + 5

= n^{3} + 5 n + {3 n}^{2} + 3 n + 6 = 6 k + 3 n (n + 1) + 6 but n (n + 1) is multiple

of 2 \Rightarrow n (n + 1) = 2 q \Rightarrow f (n + 1) = 6 k + 6 + 6 q = 6 (k + q + 1) \Rightarrow

f (n + 1) is multiple of 6