Question

what is the common difference of an ap which has its first term as 100 and the sum of its first 6 terms = 5 times the sum of its next six terms?

Answer 1

"common difference =d

Sum of n terms of an A.P. =(n/2){2a+(n−1)d}

Sum of first 6 terms =(6/2)(2a+(6−1)d} =3(2a+5d)

Hence Sum of first 12 terms =(12/2)(2a+(12−1)d} =6(2a+11d)

Hence Sum of 7th to 12th terms=Sum of first 12 terms−=Sum of first 6 terms

=6(2a+11d)−3(2a+5d) =6a+51d

SInce sum of its 1st 6 terms is 5 times the sum of the next 6 terms

→3(2a+5d)=5(6a+51d)

→6a+15d =30a+255d

→6a−30a=255d−15d= 240d

−24a =240d

and d={24a/(−240)}

Since a= 100

→d=−24×100/240 =−10

"