Question

Prove that (√2n+1 + √2n+3) is a irrational for any natural number n.

Answer 1

let (√2n+1 + √2n+3) is rational no.
then,
= (√2n+1 + √2n+3)-4 is also be rational

= (2√2n+4)- 4

= (2√2n)

= (2√2n) ÷ 2 is also be rational

= √2n ÷ n is also be rational

= √2 is also be rational
but √2 is irrational
hence (√2n+1 + √2n+3) is irrational

Answer 2

let (√2n+1 + √2n+3) is rational no.

then,

= (√2n+1 + √2n+3)-4 is also be rational

= (2√2n+4)- 4

= (2√2n)

= (2√2n) ÷ 2 is also be rational

= √2n ÷ n is also be rational

= √2 is also be rational

but √2 is irrational

hence (√2n+1 + √2n+3) is irrational