Q: In an A.P , first term = 2 & sum of first 5 terms is 1/4th of sum of next five terms. Write the equation & find d.
My solution:
$S_5$ =$\frac{1}{4}$*{$S_{10}$- $S_5$}. Q says that it is equal to sum of $\frac{1}{4}$ th of next 5 terms I.e summation of 6th , 7th , 8th , 9th , 10th term. It doesn’t not say it equals to $S_{10}$.
According to my textbook , Answer is :
$S_5$ = $\frac{1}{4}$ * $S_{10}$ .
How is this right ?
Let $a_n$ be the nth term and d be the common difference. Then according to your question \begin{align*} S_5= 5a+10d = 1/4(5a+35d) \\ 20a+40d = 5a +35d \\ a=2 \implies 40+40d = 10+35d \\ d= -6 \end{align*}
What has been written in your textbook maybe a typo and please check your calculations. It should be $S_{10}= 5S_5$.