Question

For an A.P if ${T_n} = 6n + 5$ then find ${S_n}$.

Answer 1

Hint: Here we will find the sum of first n numbers by applying summation from r=1 to r=n to the given ${T_n}$.

Complete step-by-step answer:

${S_n} = \sum\limits_{r = 1}^n {6r + 5} $
Now break the summation
$
   \Rightarrow {S_n} = \sum\limits_{r = 1}^n {6r + \sum\limits_{r = 1}^n 5 } \\
   \Rightarrow {S_n} = 6\sum\limits_{r = 1}^n r + \sum\limits_{r = 1}^n 5 \\
$
Now as you know the sum of first natural numbers is $\sum\limits_{r = 1}^n r = \dfrac{{n(n + 1)}}{2}$ and you know $\sum\limits_{r = 1}^n 5 = 5n$.
Substitute this value
$ \Rightarrow {S_n} = 6\left( {\dfrac{{n(n + 1)}}{2}} \right) + 5n = 3n(n + 1) + 5n = 3{n^2} + 8n$.
So, this is the required sum of the given A.P.

Note: In this type of question always remember the sum of the first natural number, and sum of squares of natural numbers, it will help you find out the desired answer.