Question

Match the following with their equivalents.

• A. 1+3+5+7+9+11+13
• B. 12+13
• C. $5\times 7$
• D. 6+10

Answer 1

• Every square number $n$ can be written as the sum of consecutive $n$ odd natural numbers starting from 1.

So, 1+3+5+7+9+11+13 = 7² = 49

• For any odd number 'n', $n^2=\frac{n^2-1}{2}+\frac{n^2+1}{2}$

So, for $25=5^2$,
$25=\frac{25-1}{2}+\frac{25+1}{2}=12+13$

• For any natural number 'n', $n^2-1=(n-1)(n+1)$

So, $6^2-1=(6-1)(6+1)=5\left(7\right)=35$

• Any square number can be written as sum of 2 consecutive triangular numbers.

So, $4^2=16=6+10$,
where, 6 and 10 both are triangular numbers.