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Find the sum of n terms of series 1.3.5.7, 3.5.7.9 till n?
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Find the sum of n terms of series 1.3.5.7, 3.5.7.9 till n?
Sum of Series 1.3.5.7, 3.5.7.9 till n


Introduction

The given series is a combination of two arithmetic progressions. The first progression has a common difference of 2 and starts with 1, and the second progression has a common difference of 2 and starts with 3.

Formula for the Sum of n Terms of an Arithmetic Progression

The formula for the sum of n terms of an arithmetic progression is:

Sn = n/2 * [2a + (n-1)d]

Where Sn is the sum of n terms, a is the first term, d is the common difference, and n is the number of terms.

Calculating the Sum of the First Progression

The first progression starts with 1 and has a common difference of 2. Therefore, the formula for the sum of n terms of the first progression is:

Sn1 = n/2 * [2(1) + (n-1)2]
= n/2 * [2 + 2n - 2]
= n/2 * (2n)
= n^2

Calculating the Sum of the Second Progression

The second progression starts with 3 and has a common difference of 2. Therefore, the formula for the sum of n terms of the second progression is:

Sn2 = n/2 * [2(3) + (n-1)2]
= n/2 * [6 + 2n - 2]
= n/2 * (2n + 4)
= n(n+1)

Calculating the Sum of the Entire Series

To calculate the sum of the entire series, we need to add the sum of the first progression and the sum of the second progression:

S = Sn1 + Sn2
= n^2 + n(n+1)
= n^2 + n^2 + n
= 2n^2 + n

Therefore, the sum of the series 1.3.5.7, 3.5.7.9 till n is 2n^2 + n.
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Find the sum of n terms of series 1.3.5.7, 3.5.7.9 till n?
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Find the sum of n terms of series 1.3.5.7, 3.5.7.9 till n?
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