Question

Square of any odd number n can be expressed as sum of two consecutive numbers as
$n^2=\frac{n^2-1}{2}+\frac{n^2+1}{2}$.

• A. True
• B. False

Answer 1

The correct option is A True

Let's pick a random odd number say 5
square of 5 is 25.
Given,
$n^2=\frac{n^2-1}{2}+\frac{n^2+1}{2}$...(i)
Substituting 5 in (i)
$5^2=\frac{5^2-1}{2}+\frac{5^2+1}{2}$
$25=12+\mathrm{13...}\text{(ii)}$

Let's try it with another odd number 7
square of 7 is 49.
Substituting 7 in (i)
$7^2=\frac{7^2-1}{2}+\frac{7^2+1}{2}$
$49=24+\mathrm{25...}\text{(iii)}$

From (ii) and (iii),
We can conclude that square of any odd number n can be expressed as sum of two consecutive numbers as
$n^2=\frac{n^2-1}{2}+\frac{n^2+1}{2}$.

Hence, True.