Question

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

Answer 1

Hint:
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: $S_n={\displaystyle \frac{n}{2}}[2a+(n-1)d]$. 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
$S_n={\displaystyle \frac{n}{2}}[2a+(n-1)d]$
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
$S_{10}={\displaystyle \frac{10}{2}}[2(50)+(10-1)(-4)]$
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.