Question

If the first term of an A.P. is $ a $ , second term is $ b $ and last term is $ c $ , then show that the sum of all terms is $ \dfrac{{\left( {a + c} \right)\left( {b + c - 2a} \right)}}{{2\left( {b - a} \right)}} $ .

Answer 1

Hint: To solve this question, we need to understand the concept of arithmetic progression (A.P.). Our first step will be to find the number of terms in this arithmetic progression by using the last term of the given arithmetic progression. After that using this last term in the formula for the sum of arithmetic progression and reach to the result.
Formulas used:
 $ {T_n} = a + \left( {n - 1} \right)d $ , where $ {T_n} $ is the $ {n^{th}} $ term of A.P., $ a $ is the first term of A.P. and $ d $ is the difference between two consecutive terms of A.P.
 $ {S_n} = \dfrac{n}{2}\left( {a + l} \right) $ , where, $ {S_n} $ is the sum of n terms of A.P., $ a $ is the first term of A.P. and $ d $ is the last term of A.P.

Complete step-by-step answer:
Here, we are given that the first term of an A.P. is $ a $ , second term is $ b $ and last term is $ c $ .
Therefore the difference between two consecutive terms is given by $ d = b - a $
As we know the last term of A.P., we will find the total number of terms n by using the formula $ {T_n} = a + \left( {n - 1} \right)d $
 $
  {T_n} = a + \left( {n - 1} \right)d \\
   \Rightarrow c = a + \left( {n - 1} \right)(b - a) \\
   \Rightarrow n - 1 = \dfrac{{c - a}}{{b - a}} \\
   \Rightarrow n = \dfrac{{c - a}}{{b - a}} + 1 \\
   \Rightarrow n = \dfrac{{c - a + b - a}}{{b - a}} \\
   \Rightarrow n = \dfrac{{ - 2a + b + c}}{{b - a}} \;
  $
Now, to find the sum of all terms, we will use the formula $ {S_n} = \dfrac{n}{2}\left( {a + l} \right) $ . In this we will take $ n = \dfrac{{ - 2a + b + c}}{{b - a}} $ and $ l = c $ .
 $
  {S_n} = \dfrac{n}{2}\left( {a + l} \right) \\
   \Rightarrow {S_n} = \dfrac{{ - 2a + b + c}}{{2\left( {b - a} \right)}}\left( {a + c} \right) \\
   \Rightarrow {S_n} = \dfrac{{\left( {a + c} \right)\left( {b + c - 2a} \right)}}{{2\left( {b - a} \right)}} \;
  $
Thus, the sum of all the terms of given arithmetic progression is $ \dfrac{{\left( {a + c} \right)\left( {b + c - 2a} \right)}}{{2\left( {b - a} \right)}} $ .
So, the correct answer is “ $ \dfrac{{\left( {a + c} \right)\left( {b + c - 2a} \right)}}{{2\left( {b - a} \right)}} $ .”.

Note: In this type of question, whenever the last term of the arithmetic progression is given, our first step should be to find the total number of terms in that arithmetic progression. For example, here, we have determined the number of terms in terms of a, b, and c. After that, by using this number, we have reached our final answer.