Question

Find the common difference of an AP whose first term is 1 and the sum of the first four terms is one -third to the sum of the next four terms.

Answer 1

Sum of first 4 terms =$\frac{4}{2}(2+3d)=4+6d$
Sum of first 8 terms $=\frac{8}{2}(2+7d)=8+28d.$
Hence sum of 5th to 8th term = 8 + 28d - 4 - 6d = 4 + 22d
Therefore, $4+6d=\frac{1}{3}(4+22d)$
12 + 18 d = 4 + 22d
4d = 8 or d = 2.
Check: the AP is 1, 3,,5,7,9,11,13,15,
Sum of first 4 terms = 1+3 + 5 + 7 = 16
Sum of next 4 terms = 9 + 11 + 13 + 15 = 48