Question

Find the sum of the following arithmetic progression.
$50, 46, 42,....$ to $10$ terms.

Answer 1

Here, they have given an arithmetic series and they are asking to find the arithmetic progression. So, we have a formula to find the sum of $n$ terms of an arithmetic progression that is given by: ${S_n} = \dfrac{n}{2}[2a + (n - 1)d]$. By using this formula we can get the required answer.

Complete Step by Step Solution:
In the given problem they have given an arithmetic progression, and asking us to find sum of the arithmetic progression.
In order to find the sum of $n$ terms of an arithmetic progression we have a formula, given by
${S_n} = \dfrac{n}{2}[2a + (n - 1)d]$
Where ${S_n}$ is the sum of $n$ terms.
$n =$ number of terms.
$a =$ first term of the given arithmetic progression.
$d =$ common difference of the arithmetic progression.
The series is given by $50,46,42,....$ to $10$ terms.
From the given arithmetic progression, we can say that
Number of terms $n = 10$
The first term of the given arithmetic progression i.e., $a = 50$
Now, to find the common difference that is $d$, we follow the following procedure or formula
$d = {a_2} - {a_1}$Here, ${a_1} = 50$ and ${a_2} = 46$, by substituting these values, we get
$d = 45 - 50 = - 4$
Now, by substituting the values of $a,d$ and $n$in the sum of $n$ terms of an arithmetic progression formula, we get
${S_{10}} = \dfrac{{10}}{2}[2(50) + (10 - 1)( - 4)]$
On simplifying the above equation,
$\Rightarrow {S_{10}} = (5)[100 + (9)( - 4)]$
$\Rightarrow {S_{10}} = (5)[100 - 36]$
$\Rightarrow {S_{10}} = (5)[64]$
$\Rightarrow {S_{10}} = 320$
Therefore the sum of the arithmetic progression that is $50,46,42,....$ to $10$ terms is $320$.

Note:
If you want to cross verify the answer, or if you are not able to recall the formula then we can find the entire series, that is from the given expression $50,46,42,....$ to $10$ terms, we can notice that there is a difference of $4$for every number so by using this difference first try to find the entire series then by adding these we can get the answer to cross verify.