# Consider the following statements: 1. The sum of cubes of first 20 natural numbers is 444000. 2. The sum of squares of first 20 natural numbers is 287

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Consider the following statements:

1. The sum of cubes of first 20 natural numbers is 444000.

2. The sum of squares of first 20 natural numbers is 2870.

Which of the above statements is / are correct?

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
5. None of these

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Correct Answer - Option 2 : 2 only

Concept:

Sum of cubes of natural numbers:

The sum of the cubes of first n natural numbers is given by:

$\rm \displaystyle\sum_{k=1}^{n}k^3 = \left[\dfrac{n(n+1)}{2}\right]^2$

Sum of squares of natural numbers:

The sum of squares of first n natural numbers is given by:

$\rm \displaystyle\sum_{k=1}^{n}k^2 = \left[\dfrac{n(n+1)(2n+1)}{6}\right]$

Calculation:

The sum of cubes of the first 20 natural numbers is given by:

\begin{align*} \left[\dfrac{n(n+1)}{2}\right]^2 &= \left[\dfrac{20(20+1)}{2}\right]^2\\ &= (210)^2\\ &=44100 \end{align*}

Similarly, the sum of squares of the first 20 natural numbers is given by:

\begin{align*} \left[\dfrac{n(n+1)(2n+1)}{6}\right] &= \dfrac{20(20+1)(40+1)}{6}\\ &= \dfrac{20\times 21\times41}{6}\\ &= 2870 \end{align*}

Therefore, sum of cubes of the first 20 natural numbers is 44100 and that of squares of the first 20 natural numbers is 2870.

Thus only the second statement is correct.