# 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.

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 ?

• The equation $S_5 = \frac{1}{4}S_{10}$ is false since $S_5 \neq 0$, as you can verify by solving the problem. Jan 19, 2022 at 18:10
• @N.F.Taussig Textbook solution says it’s correct. I’ll check it again. What do you think about the solution I have written ? Jan 19, 2022 at 19:50
• 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 $d$. Jan 19, 2022 at 22:31
• Have you made progress? Jan 21, 2022 at 12:06
• @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. Feb 2, 2022 at 9:49

## 1 Answer

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 $$S_{10}= 5S_5$$.

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Jan 19, 2022 at 16:42