Sum of Squares of Consecutive Integers

CAT 2017 Slot 2 · QA · Hard · Number Theory

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If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is

  1. A.

    1777

  2. B.

    1785

  3. C.

    1875

  4. D.

    1877

Answer

D

Explanation

Let the three consecutive integers be n−1,n,n+1n-1, n, n+1.

Product (n−1)n(n+1)=n3−n=15600(n-1)n(n+1) = n^3 - n = 15600.

Since n3≈15600n^3 \approx 15600, n≈156003≈25n \approx \sqrt[3]{15600} \approx 25.

Let's test n=25n = 25: 24×25×26=1560024 \times 25 \times 26 = 15600.

Sum of squares = (n−1)2+n2+(n+1)2=242+252+262=576+625+676=1877(n-1)^2 + n^2 + (n+1)^2 = 24^2 + 25^2 + 26^2 = 576 + 625 + 676 = 1877.

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