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# Find the sum of the following series.52 + 72 + 92 + … + 392

Given the the series S = 52 + 72 + 92 + … + 392

Let S1 = 12 + 32

Let S2 = 22 + 42 + 62 + .. + 382 with n = 19

Let S3 = 12 + 22 + 32 + 42 + .. + 392 with n = 39

Required sum S = S3-S2-S1

To find the sum S1 -

S1 = 1 + 9 = 10

To find the sum S2

S2 can be rewritten as S2 = 22(12 + 22 + 32 + 42 + .. + 192)

Formula to find the sum of first n squares of natural numbers is

S =  S2 = = = S2 = 9880

To find the sum S3 -

Formula to find the sum of first n squares of natural numbers is

S =  S3 = = = 20540

Required sum S = S3-S2-S1

S = 20540-9880-10 = 10650 The sum S = 52 + 72 + 92 + … + 392 = 10650

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