First slide

Series of natural numbers

Question

$Find the sum of the series 31^{3} + 32^{3} + \dots . + 50^{3}$

Moderate

A

1409400
B

1309400
C

1509400
D

1609400

Solution

$S = (1^{3} + 2^{3} + .... + 50^{3}) - (1^{3} + 2^{3} + ... + 30^{3})$
$\begin{array}{l} = {(\frac{50 \times 51}{2})}^{2} - {(\frac{30 \times 31}{2})}^{2} \\ = \frac{1}{4} (50 \times 51 - 30 \times 31) (50 \times 51 + 30 \times 31) = 1409400 \end{array}$

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