Statement 1 : The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 +12 + 16) + ...... + (361 + 380 + 400) is 8000. Statement 2 : \sum_{k = 1}^{n} \left(\right. k^{3} - \left(\left(\right. k - 1 \left.\right)\right)^{3} \left.\right) = n^{3}, for any natural number n.

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Mathematics

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Question

Statement 1 : The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 +12 + 16) + ...... + (361 + 380 + 400) is 8000.

Statement 2 : $\sum_{k = 1}^{n} (k^{3} - {(k - 1)}^{3}) = n^{3}$ , for any natural number n.

Statement 1 is true, Statement 2 is false.

Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 2.

Statement 1 is false, Statement 2 is true.

Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

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S 1: 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 +12 + 16) + ...... + (361 + 380 + 400) Clearly number of terms in sequence are 20.

S 2 : $\sum_{k = 1}^{n} (k^{3} - {(k - 1)}^{3}) = n^{3}$ is true.

$∴$ The sum of given series for 20 terms is (20)³ = 8000.