Answer :

Given the the series S = 163 + 173 + … + 353

Let S1 = 13 + 23 + 33 + … + 353 with n = 35


Let S2 = 13 + 23 + 33 + .. + 153 with n = 15


Required sum S = S1-S2


To find the sum S1 -


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


S =


S1 =


=


S1 = 396900


To find the sum S2 -


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


S =


S2 =


=


S2 = 14400


Required sum S = S1-S2


S = 396900-14400 = 382500


The sum S = 163 + 173 + … + 353 = 382500


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