- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# The sum of the first $ n $ terms of an AP whose first term is 8 and the common difference is 20 is equal to the sum of first $ 2 n $ terms of another AP whose first term is $ -30 $ and the common difference is 8 . Find $ n $.

Given:

The sum of first $n$ terms of an A.P. whose first term is 8 and the common difference is 20 is equal to the sum of first $2n$ terms of another A.P. whose first term is $-30$ and common difference is 8.

To do:

We have to find the value of $n$.

Solution:

Let the first A.P. be $A_1$ and the second A.P. be $A_2$.

First term of the first A.P. $a = 8$

Common difference of the first A.P. $d = 20$

Let the number of terms in first A.P. be $n$

Sum of first $n$ terms of an A.P.,

$S_n=\frac{n}{2}[2a+(n-1)d]$

$=\frac{n}{2}[2\times8+(n-1)20]$

$=\frac{n}{2}(16+20n-20)$

$=\frac{n}{2}(20 n-4)$

$=n(10 n-2)$......(i)

First term of the second A.P. \( \left(a^{\prime}\right)=-30 \)

Common difference of the second A.P. \( \left(d^{\prime}\right)=8 \)

Therefore,

Sum of first \( 2 n \) terms of second A.P.,

\( \mathrm{S}_{2 n}=\frac{2 n}{2}\left[2 a^{\prime}+(2 n-1) d^{\prime}\right] \)

\( =n[2(-30)+(2 n-1) 8] \)

\( =n[-60+16 n-8] \)

\( =n[16 n-68] \)......(ii)

According to the question,

Sum of first \( n \) terms of the first \( A P \) \( = \) Sum of first \( 2 n \) terms of the second A.P.

\( \Rightarrow \mathrm{S}_{n}=\mathrm{S}_{2 n} \)

\( \Rightarrow n(10 n-2)=n(16 n-68) \)

\( \Rightarrow n[(16 n-68)-(10 n-2)]=0 \)

\( \Rightarrow n(16 n-68-10 n+2)=0 \)

\( \Rightarrow n(6 n-66)=0 \)

\( 6n=66 \) or \( n=0 \) which is not possible

Therefore,

$n=11$

Hence, the required value of \( n \) is 11.

- Related Articles
- The sum of first $n$ terms of an A.P. whose first term is 8 and the common difference is 20 is equal to the sum of first $2n$ terms of another A.P. whose first term is $-30$ and common difference is 8. Find $n$.
- Find The first four terms of an AP, whose first term is $–2$ and the common difference is $–2$.
- In an A.P. the first term is 8, nth term is 33 and the sum to first n terms is 123. Find n and d, the common differences.
- In an A.P., the first term is 22, nth term is $-11$ and the sum to first n terms is 66. Find n and d, the common difference.
- If the sum of the first n terms of an AP is $4n - n^2$, what is the first term (that is $S_1$)? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.
- The sum of first 7 terms of an AP is 49 and that of 17 terms is 289. Find the sum of first n terms .
- The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
- The first term of an AP is \( -5 \) and the last term is 45 . If the sum of the terms of the AP is 120 , then find the number of terms and the common difference.
- If the sum of the first $n$ terms of an A.P. is $4n – n^2$, what is the first term? What is the sum of first two terms? What is the second term? Similarly, find the third, the tenth and the $n$th terms.
- If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first $n$ terms.
- In an A.P., if the first term is 22, the common difference is $-4$ and the sum to n terms is 64, find $n$.
- The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP?
- The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235 , find the sum of its first twenty terms.
- If the sum of first p terms of an A.P., is $ap^{2} +bp$, find its common difference.
- The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.