Question

Find the sum of
5² +10² +15² + ... +105²​

Answer 1

Answer:

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Answer 2

5² + 10² + 15² + . . . + 105² = 82775

Given :

5² + 10² + 15² + . . . + 105²

To find :

The sum

Formula :

$\displaystyle \sf{ {1}^{2} + {2}^{2} + {3}^{2} + ... + {n}^{2} = \frac{n(n + 1)(2n + 1)}{6} }$

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

5² + 10² + 15² + . . . + 105²

Step 2 of 2 :

Find the value of the expression

$\displaystyle \sf{ {5}^{2} + {10}^{2} + {15}^{2} + ... + {105}^{2} }$

$\displaystyle \sf{ = {(5 \times 1)}^{2} + {(5 \times 2)}^{2} + {(5 \times 3)}^{2} + ... + {(5 \times 21)}^{2} }$

$\displaystyle \sf{ = {5}^{2} \bigg[ {1}^{2} + {2}^{2} + {3}^{2} + ... + {21}^{2} \bigg] }$

$\displaystyle \sf{ = {5}^{2} \times \frac{21 \times (21 + 1) \times (42 + 1)}{6} }$

$\displaystyle \sf{ = 25 \times \frac{21 \times 22 \times 43}{6} }$

$\displaystyle \sf{ = 25 \times 7 \times 11 \times 43 }$

$\sf = 82775$

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