Question

If n is a positive integer, what is the smallest value of n such that

(n+10) + (n+11) + (n+12) + ......+ (n+58) is a perfect square?

• A. 2
• B. 4
• C. 3
• D. 1

Answer 1

The correct option is A

2

This is an arithmetic sequence with 49 terms, and middle term (n + 34), so the sum is equal to 49(n + 34). Since 49 is $7^2$, which is already a perfect square, we need the smallest value of n such that (n + 34) is a perfect square. We see that for n=2, n+34 is a perfect square (since $36=6^2$).