Question

What constant should we add to `n^2+n+_...` to make it a perfect square?

Answer 1

The proper constant to add is `1/4` and `n^2+n+1/4=n^2+2xxnxx1/2+(1/2)^2=(n+1/2)^2`

Explanation:

Compare `n^2+n+_...` with LHS of `a^2+2xxaxxb+b^2=(a+b)^2`

It is apparent that `n=a`

If it so, how can we write `n` as `2xxaxxb`

well we need `2`, then as `a` follows we should have `n`

i.e. we have `2xxnxx...`

if this is equal to second term in `n^2+n+_...`

we must have `2xxnxx1/2` i.e. `1/2` is our `b`

and hece third term in trinomial has to be `(1/2)^=1/4`

Hence to complete perfect square trinomial we have

`n^2+2xxnxx1/2+(1/2)^2` and as `a=n` and `b=1/2`

`n^2+n+1/4=n^2+2xxnxx1/2+(1/2)^2=(n+1/2)^2`