Question

The sum of $18$ consecutive natural numbers is a perfect square. What is the smallest possible value of this sum?

• A. $169$
• B. $225$
• C. $289$
• D. $441$

Answer 1

Answer: (B)

$225$

Let the numbers be
$a,a+1,a+2,.....a+17$
This is an A.P with
First term $=a$
Common difference $d=1$
Terms $n=18$
Sum of the terms
$S_n==\frac{18}{2}(2a+17)$
$=9(2a+17)$
$=18a+153$
For $a=1,S_n=171$
$a=1,S_n=189$
$a=3,S_n=207$
$a=4,S_n=225$
$={15}^2$ (A perfect square)