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Find the sum of series 1, 3, 5, 7, 9,…… up to n terms?

Answer
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Hint: We can see that the given series in the problem follows an Arithmetic Progression (AP). We find the nth term of the series by using the first term and common difference of the given series. Once, we find the nth terms, we use the sum of n terms of Arithmetic Progression (AP) to get the desired result.

Complete step-by-step solution:
According to the problem, we have a series given as 1, 3, 5, 7, 9,……. We need to find the sum of the elements in this series up to n terms.
We know that an Arithmetic Progression (AP) is of form a, a+d, a+2d,…….., where ‘a’ is known as the first term and ‘d’ is known as common difference.
We can see that the 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’.
Let us find the nth term for the given series. We know that nth term of an Arithmetic Progression (AP) is defined as Tn=a+(n1)d.
So, we have nth term for the given series as Tn=1+(n1)2.
Tn=1+2n2.
Tn=2n1 ---(1).
Now let us find the sum of n terms of the given series. We know that sum of n terms of an Arithmetic Progression (AP) is defined as Sn=n2×(a+Tn).
So, we have sum of n terms of the given series as Sn=n2×(1+2n1).
Sn=n2×(2n).
Sn=n×n.
Sn=n2.
We have found the sum of n terms of the series 1, 3, 5, 7, …… is n2.
The sum of series 1, 3, 5, 7, 9,…… up to n terms is n2.

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

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