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 ?

  • $\begingroup$ The equation $S_5 = \frac{1}{4}S_{10}$ is false since $S_5 \neq 0$, as you can verify by solving the problem. $\endgroup$ Jan 19, 2022 at 18:10
  • $\begingroup$ @N.F.Taussig Textbook solution says it’s correct. I’ll check it again. What do you think about the solution I have written ? $\endgroup$
    – S.M.T
    Jan 19, 2022 at 19:50
  • $\begingroup$ The equation $S_5 = \frac{1}{4}(S_{10} - S_5)$ is correct. If you substitute $2$ for $a_1$ in the formulas $a_n = a_1 + d(n - 1)$ and $S_n = \frac{n(a_1 + a_n)}{2}$, you can apply the equation $S_5 = \frac{1}{4}(S_{10} - S_5)$ to find $d$. $\endgroup$ Jan 19, 2022 at 22:31
  • $\begingroup$ Have you made progress? $\endgroup$ Jan 21, 2022 at 12:06
  • 1
    $\begingroup$ @S.M.T I see! I had no clue about that, if that's true then it's unfortunate because for classes 6 and below, NCERT has fantastic content. Actually, it's kind of unfortunate that NCERT needs to publish books for 11th and 12th standard kids. I'd rather they gave it a good go in the children segment and left the competitive segment to the boards and other private institutions. I was under the impression that those institutions do a worse job, but if that's not true then I would make such a suggestion to NCERT. Anyway, people at NCERT don't strike me as wanting to write competitive books. $\endgroup$ Feb 2, 2022 at 9:49

1 Answer 1


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 $S_{10}= 5S_5$.

  • $\begingroup$ As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Jan 19, 2022 at 16:42

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