Question

Find the sum of the following series without actually adding it.
$1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21$

Answer 1

We can see the given series in the pattern follows an Arithmetic progression (AP). We need to find the sum of the given series by using the formula ${n^2}$. We use the sum of the given terms that is of the form of series.
We use the sum of Arithmetic progression (AP) to get the required result.

Complete step-by-step answer:
According to the problem, we have a series given as $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21$
We need to find the sum of the elements in the series up to $21$
Here the arithmetic Progression (AP) is of form $a, a + 2d,a + 3d.....$ where $'a'$ is known as the first term and $'d'$ is known as common difference.
We can see that the given series has a difference between any two consecutive numbers is $2$
We can see that the given series are in AP (Arithmetic Progression) with the first term $1$ and common difference $2$
We are proceeding the way we are doing,
Here, total consecutive odd numbers $= 11$
∴ $n = 11$
Sum of the consecutive odd numbers $= {n^2}$
$= {11^2}$
We need to multiply ${\text{11 by 11}}$ we get,
$= 121$
Hence we have found the sum of the asked terms.

Therefore the sum of the above series terms is $121$

Note: We can also use the sum of $'n'$ natural numbers to find the sum of the given series after calculating the ${n^{th}}$ term of the series.
Whenever we see a problem following Arithmetic Progression (AP), we make use of the term ${n^{th}}$ term. Similarly, we can expect problems to find the sum of the even numbers up-to n-terms