Question
Quantity-I: The total sum of all
perfect square numbers in the range from 1 to 1000. Quantity-II: The total sum of all perfect cube numbers in the range from 1 to 2500. In the question, two Quantities I and II are given. You have to solve both the Quantity to establish the correct relation between Quantity-I and Quantity-II and choose the correct option.Solution
ATQ, Quantity I: Largest perfect square before 1000 = 312 = 961 So, required sum = sum of squares of all natural numbers up to 31. So, required sum = {n(n + 1)(2n + 1)} ÷ 6 = (31 × 32 × 63)/6 = 10416 So, Quantity I = 10416 Quantity II Largest perfect cube before 2500 = 133 = 2197 So, required sum = [n2 (n + 1)2 ]/4 = {(13)2 × (14)2 }/4 = 8281 So, Quantity II = 8281 So, Quantity I > Quantity II
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