If sum of first 6 terms of an AP is 36 and that of the first 16 terms is 256 , find the sum of first 10 terms.
Given:
The sum of first 6 terms of an \( \mathrm{AP} \) is 36 and the sum of its first 16 terms is 256.
To do:
We have to find the sum of its first $10$ terms.
Solution:
Let the first term be $a$ and the common differnce be $d$.
We know that,
Sum of $n$ terms$ S_{n} =\frac{n}{2}(2a+(n-1)d)$
$S_{6}=\frac{6}{2}[2(a)+(6-1)d]$
$36=3(2a+5d)$
$12=2a+5d$
$2a=12-5d$......(i)
$S_{16}=\frac{16}{2}[2(a)+(16-1)d]$
$256=8(2a+15d)$
$32=2a+15d$
$12-5d+15d=32$ (From (i))
$10d=32-12$
$d=\frac{20}{10}$
$d=2$
This implies,
$2a=12-5(2)$
$2a=12-10$
$a=\frac{2}{2}$
$a=1$
The sum of $10$ terms $S_{10}=\frac{10}{2}[2(1)+(10-1)2]$
$=5[2+9(2)]$
$=5(2+18)$
$=5(20)$
$=100$
Hence, the sum of $10$ terms is $100$.
Related Articles
- The sum of first 6 terms of an \( \mathrm{AP} \) is 36 and the sum of its first 16 terms is 256. Find the sum of first 10 terms of this \( A P \).
- 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.
- Find the sum of first $16$ terms of the AP: $10,\ 6,\ 2\ .......$
- 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.
- 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 .
- If the sum of first four terms of an A.P. is 40 and that of first 14 terms is 280. Find the sum of its first n terms.
- In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P.
- If 12th term of an A.P. is $-13$ and the sum of the first four terms is 24, what is the sum of first 10 terms?
- The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.
- 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 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 \( n \).
- Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
- If the sum of first p terms of an A.P., is $ap^{2} +bp$, find its common difference.
- Find the sum of the first 11 terms of the A.P.: $2, 6, 10, 14,…..$
- If the $6^{th}$ term of an A.S is 64, find the sum of the first 11 terms.
Kickstart Your Career
Get certified by completing the course
Get Started