Find the number of perfect squares in the given series 2013, 2020,…

The given series is an AP with common difference of 7. So the terms in the above series are in the form of 2013 + 7k. We have to find the perfect squares in this format in the given series.

Given that 44^2 = 1936.

Shortcut: To find the next perfect square, add 45th odd number to 44^2.

So 45^2 = 1936 + (2 x 45 -1) = 2025

46^2 = 2025 + (2 x 46 - 1) = 2116

47^2 = 2116 + (2 x 47 - 1) = 2209

Now subtract 2013 from the above numbers and divide by 7. Only 2209 is in the format of 2013 + 7k. One number satisfies.