# If the sum of the first $n$ terms of an AP is $\frac{1}{2}(3n^2+7n)$ then find its $n^{th}$ term. Hence, find its $20^{th}$ term.

**Given: **the sum of the first $n$ terms of an AP is $\frac{1}{2}(3n^2+7n)$

**To do: **To find its $n^{th}$ term and its $20^{th}$ term.

**Solution:**

$S_n=\frac{1}{2}( 3n^2+7n)$

$S_1=\frac{1}{2}(3+7)=5$

$S_2=\frac{1}{2}(3\times2^2+7\times2)=\frac{26}{2}=13$

We know

$S_1=a_1=5$

$S_2=a_1+a_2=13$

$\Rightarrow S_2-S_1=a_1+a_2-a_1$

$\Rightarrow 13-5=a_2$

$\Rightarrow a_2=8$

And we also know that common difference $d=a_2-a_1$

$\Rightarrow d=8-5=3$

$n^{th}$ term of AP $=a_n=5+( n-1)3$

$a_n= 2+3n$

Therefore $20^{th}$ term $=a_{20}=2+3(20)=62$

Hence $20^{th}$ term of AP is $62$.

- Related Articles
- If the sum of first $n$ terms of an A.P. is $\frac{1}{2}(3n^2 + 7n)$, then find its $n$th term. Hence write its 20th term.
- If the sum of first n terms of an AP is $\frac{1}{2}[3n^2+7n]$ then find nth term and hence, write its 20th term.
- If the $n^{th}$ term of an AP is $\frac{3+n}{4}$, then find its $8^{th}$ term.
- If the $10^{th}$ term of an A.P is $\frac{1}{20}$ and its $20^{th}$ term is $\frac{1}{10}$, then find the sum of its first $200$ terms.
- The $14^{th}$ term of an A.P. is twice its $8^{th}$ term. If its $6^{th}$ term is $-8$, then find the sum of its first $20$ terms.
- If $m$ times the $m^{th}$ term of an AP is equal to $n$ times its $n^{th}$ term. find the $( m+n)^{th}$ term of the AP.
- In an A.P., the sum of first $n$ terms is $\frac{3n^2}{2}+\frac{13}{2}n$. Find its 25th term.
- If the sum of first $n$ terms of an A.P. is $n^2$, then find its 10th term.
- If the 10th term of an A.P. is 21 and the sum of its first ten terms is 120, find its $n$th term.
- The sum of first n terms of an A.P. is $3n^{2} +4n$. Find the $25^{th}$ term of this A.P.
- The sum of first $n$ terms of an A.P. is $5n^2 + 3n$. If its $m$th term is 168, find the value of $m$. Also, find the 20th term of this A.P.
- The $m^{th}$ term of an arithmetic progression is $x$ and the $n^{th}$ term is $y$. Then find the sum of the first $( m+n)$ terms.
- The sum of first $m$ terms of an A.P. is $4m^2 – m$. If its $n$th term is 107, find the value of $n$. Also, find the 21st term of this A.P.
- Find the $5^{th}$ term of an A.P. of $n$ terms whose sum is $n^2−2n$.
- 12th term of an \( \mathrm{AP} \) is 4 and its 20th term is \( -20 \). Find the \( n \) th term of that \( A P \).

##### Kickstart Your Career

Get certified by completing the course

Get Started