Question

Given that $S_n+1+3+5+....n$ terms. $p,q,r$ are positive integers such that $S_p+S_q=S_r$. The smallest value of p + q + r is

• A. $12$
• B. $10$
• C. $24$
• D. $18$

Answer 1

The correct option is A $12$

$S_n=1+3+5+......nterms$

The given series is in A.P.

First term, $a_1=1$

Common difference, $d=a_2-a_1=3-1=2$

Sum of first $n$ terms is,

$S_n=\frac{n}{2}\{2a+(n-1\left)d\right\}$

$=\frac{n}{2}(2+2n-2)$

$=n^2$

$S_p+S_q=S_r$

$p^2+q^2=r^2$

Also, $p,q,r$ are positive integers.

So, for smallest value of $p+q+r$

$p=3$ , $q=4$ , $r=5$

Therefore, smallest value of $p+q+r$ is $12$