Question

If 1 is the last term of the finite A.P., then write the formula to find the sum of all terms of the A.P.

Answer 1

Hint: An A.P. is a sequence of number having equal difference between any of its two consecutive terms, for example:
$a,a+d,a+2d,.................a+\left( n-1 \right)d$ is an A.P. which has $n$ terms and a common difference $d$ between any two consecutive terms. You can also see that its first term is $a$ and it's ${{n}^{th}}$ term is given by:
${{a}_{n}}=a+\left( n-1 \right)d$
Sum of all the $n$ terms of A.P. is denoted by ${{S}_{n}}$ and is given by
${{S}_{n}}=\dfrac{n}{2}\left( a+{{a}_{n}} \right)$

Complete step by step answer:
According to the question, the last term of a finite A.P. is 1 i.e.
${{a}_{n}}=1$
We know that ${{a}_{n}}=a+\left( n-1 \right)d$
$\begin{align}
  & \Rightarrow a+\left( n-1 \right)d=1 \\
 & \Rightarrow a=1-\left( n-1 \right)d\text{ }..............\text{(1)} \\
\end{align}$
Therefore, the sum of all the terms of this A.P will be given by
${{S}_{n}}=\dfrac{n}{2}\left( a+{{a}_{n}} \right)$
Taking the value of $a$ from the equation (1) and putting it in the above equation, we get
$\Rightarrow {{S}_{n}}=\dfrac{n}{2}\left( 1-\left( n-1 \right)d+{{a}_{n}} \right)$
Now putting ${{a}_{n}}=1$ in the above equation, we get
$\begin{align}
  & \Rightarrow {{S}_{n}}=\dfrac{n}{2}\left( 1-\left( n-1 \right)d+1 \right) \\
 & \Rightarrow {{S}_{n}}=\dfrac{n}{2}\left( 2-\left( n-1 \right)d \right) \\
\end{align}$

Therefore, the formula to find the sum of a finite A.P whose last term is 1 comes out to be
${{S}_{n}}=\dfrac{n}{2}\left( 2-\left( n-1 \right)d \right)$

Note: You might think of assuming some random value of $n$ and $d$ but you have to focus on what is being asked to you. You are being asked a formula in this question and that’s why there will be some unknown variables in your formula and here you don’t have to come up with a numerical value. The one thing you can see here is that the general expression of ${{n}^{th}}$ term and formula of sum of all terms in an A.P. is very important and you should always keep it in your mind in order to tackle these question easily.