Question

An A.P. consists of 31 terms. If 16th term is n, find the sum of all terms of the A.P.

Answer 1

Hint: We will assume the first term and common difference of the A.P. as variable. Then we will use the formula of ${{n}^{th}}$ term of A.P. and the sum of first n terms of A.P. which is as follows:
${{n}^{th}}$ term of A.P. $={{T}_{n}}=a+(n-1)d$
Sum of first n terms $=\dfrac{n}{2}\left[ 2a+(n-1)d \right]$, where a and d are the first term and the common difference respectively of the A.P.

Complete step-by-step answer:
We have been given that an A.P. consists of 31 terms,
$\Rightarrow$ n = 31
Also, we have been given that 16th term of the A.P. is n.
Let us assume the first term and common difference to be a and d respectively of the A.P.
We know that the ${{n}^{th}}$ term is given by,

& \Rightarrow {{T}_{n}}=a+(n-1)d \\\ & \Rightarrow {{T}_{16}}=a+(16-1)d \\\ & \Rightarrow {{T}_{16}}=a+15d \\\ & \Rightarrow n=a+15d....(1) \\\ \end{aligned}$$ We also know that the sum of the first ‘n’ terms of an A.P. is equal to ‘n’ divided by 2 times the sum of twice the first term and product of the common difference and n minus 1. $$\Rightarrow {{S}_{n}}=\dfrac{n}{2}\left[ 2a+(n-1)d \right]$$ So the sum of the first 31 terms is $${{S}_{31}}$$. $$\Rightarrow {{S}_{31}}=\dfrac{31}{2}\left[ 2a+(31-1)d \right]=\dfrac{31}{2}\left[ 2a+30d \right]$$ Taking 2 as common, we get as follows: $$\Rightarrow {{S}_{31}}=\dfrac{31}{2}\times 2\left[ a+15d \right]$$ Now, by substituting the value of $$\left[ a+15d \right]$$ from the equation (1), we get as follows: $$\begin{aligned} & \Rightarrow {{S}_{31}}=\dfrac{31}{2}\times 2\times n \\\ & \Rightarrow {{S}_{31}}=31n \\\ \end{aligned}$$ Hence the sum of all the terms of the A.P.is equal to 31n. Note: Take care of the sign mistakes while calculation of $${{T}_{n}}$$ and $${{S}_{n}}$$ of an A.P. Also, remember the formulas of $${{T}_{n}}$$ and $${{S}_{n}}$$ of an A.P. as it will help in the problems related with it. If we make a mistake in writing the formula of either of these, we will get the wrong answer.