Question

if the sum of first n terms of an Ap is 3n^2+2n the n th term is​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

Sum of first n terms of an AP series is

$\rm \: S_n = {3n}^{2} + 2n$

Now, we know that,

$\rm \: S_n = a_1 + a_2 + a_3 + - - - + a_{n - 1} + a_n$

can be further rewritten as

$\rm \: S_n = S_{n - 1} + a_n$

$\rm\implies \:a_n \: = \: S_n \: - \: S_{n - 1}$

Now, we have

$\rm \: S_n = {3n}^{2} + 2n$

So,

$\rm \: S_{n - 1} = {3(n - 1)}^{2} + 2(n - 1)$

$\rm \: = \:3( {n}^{2} + 1 - 2n) + 2n - 2$

$\rm \: = \:3{n}^{2} + 3 - 6n + 2n - 2$

$\rm \: = \:3{n}^{2} - 4n + 1$

So, on substituting the values in

$\rm \: a_n \: = \: S_n \: - \: S_{n - 1}$

we get

$\rm \: a_n = {3n}^{2} + 2n - ( {3n}^{2} - 4n + 1)$

$\rm \: a_n = {3n}^{2} + 2n - {3n}^{2} + 4n - 1$

$\rm\implies \:a_n \: = \: 6n - 1$

$\rule{190pt}{2pt}$

Additional Information :-

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

aₙ is the nᵗʰ term.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of AP.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.