Answer :

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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