Question

If Sn, the sum of first n terms of an A. P., is given by Sn = $5{n^2} + 3n$, then find its ${n^{th}}$ term.

Answer 1

Hint: To find the nth term ${a_n}$, we will use the formula of difference of sums i.e.,
an = Sn – Sn-1 where, Sn is the sum of first n terms of the A. P. and Sn-1 is the sum of the first (n – 1) terms of the A. P.

Complete step-by-step answer:
 We are given the sum of first n terms of an A. P. as: Sn = $5{n^2} + 3n$
We need to calculate the ${n^{th}}$term of this A. P.
We know the formula of the difference of sums of an A. P. is given by: ${a_n} = {S_n} - {S_{n - 1}}$ – (1)
Now, for calculating the value of Sn-1, we can say that Sn is true for first n terms then we know that
 n > n – 1 , therefore, it will be true for first n – 1 terms as well.
$\therefore {S_{n - 1}} = 5{\left( {n - 1} \right)^2} + 3\left( {n - 1} \right)$
Substituting the values of Sn and Sn-1 in equation (1), we get
\[
   \Rightarrow {a_n} = {S_n} - {S_{n - 1}} \\
   \Rightarrow {a_n} = 5{n^2} + 3n - \left\{ {5{{\left( {n - 1} \right)}^2} + 3\left( {n - 1} \right)} \right\} \\
 \]
Upon simplification, we get
$
   \Rightarrow {a_n} = 5{n^2} + 3n - \left\{ {5\left( {{n^2} - 2n + 1} \right) + 3n - 3} \right\} \\
   \Rightarrow {a_n} = 5{n^2} + 3n - 5{n^2} + 10n - 5 - 3n + 3 \\
   \Rightarrow {a_n} = 10n - 2 \\
 $
Therefore, the nth term of the A. P., whose sum of first n terms is $5{n^2} + 3n$, is found to be 10n – 2.

Note: In such questions, selecting the formula or method to proceed with to calculate the ${n^{th}}$ term of the A. P is important. You may go wrong while calculating the sum of first n – 1 terms of the A. P. since it is not provided and you have to show that the sum of the terms will be true for n – 1 terms.