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Q. Calculate the sum of the squares of first 100 terms of an AP , given that the sum of the first 100 terms is -1. and that the sum of 2nd , 4th, ......., 100th term is 1.
Asked by Shrivatsa | 22 Nov, 2018, 05:36: PM
In A.P, sum of 100 terms is -1, (100/2) [ 2a +99d ] = 50 [ 2a+99d ] = -1 .................(1)
where a is the first term and d is common difference. eqn.(1) can be rewritten as  [ 2a + 99d ] = (-1/50) ...................(2)

Similarly if sum of 2nd, 4th, 6th .......100th terms are -1 , then (50/2) [ 2(a+d) + 49×2d ] = 1  or 2a+100d = (1/25) ...............(3)

if we solve eqns. (2) and (3), we get d = 3/50  and a = -149/50

we can write the nth term of A.P. as, tn = (-149/50)+(n-1)(3/50)

Hence the required sum = .....................(4)
In eqn.(4) we get sum of integers  and sum of squares of integers from 1 to 100.

Hence the required sum as given in eqn.(4) is written as
further simplification can done by the student
Answered by Thiyagarajan K | 23 Nov, 2018, 10:02: PM

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