Question

The sum of the arithmetic series:
$5 + 11 + 17 + .... + 95$ is

Answer 1

Hint: The sum of an arithmetic series is given by the formula
$ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$
Where, $n = $ Number of terms, $a = $first term and $d = $common difference.
Here, we have all the values required except n. To find the value of n, we will use the formula
$ \Rightarrow {a_n} = a + \left( {n - 1} \right)d$
Where, we can substitute ${a_n}$ as last term or first term.

Complete step-by-step answer:
In this question, we are given an arithmetic series with the last term known and we need to find the sum of this arithmetic series.
The given arithmetic series is: $5 + 11 + 17 + .... + 95$
Now, first of all let us see what arithmetic series is.
An arithmetic series with the last term n is the sum of all the elements in the series from the first term till the last term n. It is denoted by ${S_n}$. The formula for finding the sum of arithmetic series is given by
$ \Rightarrow {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ - - - - - - - - - - - - - (1)
Where, $n = $ Number of terms, $a = $first term and $d = $common difference.
Here, in our given series,
$
   \Rightarrow a = 5 \\
   \Rightarrow l = 95 \\
   \Rightarrow n = ? \\
   \Rightarrow d = \left( {11 - 5} \right) = 6 \;
 $
Now, to find the value of n we will use the formula
$
   \Rightarrow {a_n} = a + \left( {n - 1} \right)d \\
   \Rightarrow 95 = 5 + \left( {n - 1} \right)6 \\
   \Rightarrow 95 - 5 = \left( {n - 1} \right)6 \\
   \Rightarrow \dfrac{{90}}{6} = \left( {n - 1} \right) \\
   \Rightarrow n - 1 = 15 \\
   \Rightarrow n = 15 + 1 \\
   \Rightarrow n = 16 \;
 $
Hemce, there are a total 16 terms in the given series.
Therefore, putting these values in the equation (1), we get
$
   \Rightarrow {S_{16}} = \dfrac{{16}}{2}\left( {2\left( 5 \right) + \left( {16 - 1} \right)6} \right) \\
   \Rightarrow {S_{16}} = 8\left( {10 + 90} \right) \\
   \Rightarrow {S_{16}} = 8\left( {100} \right) \\
   \Rightarrow {S_{16}} = 800 \;
 $
Therefore, the sum of the given series $5 + 11 + 17 + .... + 95 = 800$.
So, the correct answer is “800”.

Note: The sum of arithmetic series whose first term and last term is known can also be found using the below formula.
$ \Rightarrow {S_n} = \dfrac{n}{2}\left( {{a_1} + {a_n}} \right)$
Where, $n = $Total number of terms, ${a_1} = $ first term and ${a_n} = $Last term.
Therefore, we get
$
   \Rightarrow {S_n} = \dfrac{{16}}{2}\left( {5 + 95} \right) \\
   \Rightarrow {S_n} = 8\left( {100} \right) \\
   \Rightarrow {S_n} = 800 \;
 $