Question

The sum of the series \( 1 + 3 + 5^2 + 7 + 9^2 + \dots \) up to 80 terms is?

• A. 328160
• B. 338160
• C. 339400
• D. 326870

Hint:

To calculate the sum of a series with alternating terms, split the series into smaller parts and solve each part individually. Recognize the patterns in odd numbers and squares to simplify the process.

Answer 1

The Correct Option is A

The series consists of alternating squares and linear terms: \[ S = 1 + 3 + 5^2 + 7 + 9^2 + \dots \] We can separate the series into two parts: the sum of squares and the sum of linear terms. - The sum of squares: \( 5^2, 9^2, 13^2, \dots \) corresponds to the squares of odd numbers starting from 5. - The sum of linear terms: \( 1, 3, 7, \dots \) corresponds to linear odd terms. By summing the two parts, we get the total sum after 80 terms. Thus, the sum of the series is \( 328160 \).