Prove without using induction that nn 1 is an even number fo

Prove, without using induction, that n(n +1) is an even number for every nonnegative integer n. Prove, using induction, that n(n + 1) Ls an even number for every nonnegative integer n.

Solution

a) consider any integer n,then n+1 is either odd or even depends on the integer we have taken for n.

eg: 2, 2+1=3 is an odd integer as 2 is a even number.consider 3 then 3+1 = 4 is even

therefore the product of n and n+1 is an even number since odd * even = even * odd = even.

b) 1.2 = 2 , for n=1,

for n=2, 2.3 = 6,

for n=3, 3.4 = 12

and so on for n= k, k.k+1 is an even .

we have to prove for k+1 then by induction it is proved.

k+1. K+1 + 1 = (k+1)(k+2) is an even number whatever K value may be.