Question:easy

What is the difference between the sum of the cubes and that of the sum of the square of first twenty natural numbers?

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Use the standard formulas for sum of cubes and sum of squares of the first n natural numbers with n = 20, then subtract.
Updated On: Jul 21, 2026
  • 21290
  • 28630
  • 36340
  • 41230
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The Correct Option is D

Solution and Explanation

Instead of computing the two sums separately and subtracting them, we can build one combined formula for the difference and then plug in n = 20 directly.

The sum of cubes of the first n natural numbers is \(\left(\frac{n(n+1)}{2}\right)^2\), and the sum of squares is \(\frac{n(n+1)(2n+1)}{6}\). Subtracting the second from the first and pulling out the common factor n(n+1):

\[ \sum k^3 - \sum k^2 = n(n+1)\left[\frac{n(n+1)}{4} - \frac{2n+1}{6}\right] \]

Taking the terms inside the bracket to a common denominator of 12:

\[ \frac{n(n+1)}{4} - \frac{2n+1}{6} = \frac{3n(n+1) - 2(2n+1)}{12} = \frac{3n^2 - n - 2}{12} \]

The quadratic \(3n^2 - n - 2\) factors as \((3n+2)(n-1)\), so the whole difference simplifies to a single clean formula:

\[ \sum k^3 - \sum k^2 = \frac{n(n+1)(n-1)(3n+2)}{12} \]

Now put n = 20 into this formula:

\[ \frac{20 \times 21 \times 19 \times 62}{12} \]

Multiply the numerator step by step: 20 x 21 = 420, then 420 x 19 = 7980, then 7980 x 62 = 4,94,760. Dividing this by 12 gives 41,230.

Let's summarize:

  • The difference between the sum of cubes and the sum of squares of the first n natural numbers always equals n(n+1)(n-1)(3n+2)/12.
  • Plugging n = 20 into this single formula gives the answer directly, without working out the two sums on their own.

So the required difference is 41230.

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