The sum of the series \( 1 + 3 + 5^2 + 7 + 9^2 + \dots \) up to 80 terms is?
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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.
The series comprises alternating square and linear terms:\[S = 1 + 3 + 5^2 + 7 + 9^2 + \dots\]This series can be decomposed into two separate series:- A series of squares: \( 5^2, 9^2, 13^2, \dots \), which are squares of odd numbers starting from 5.- A series of linear terms: \( 1, 3, 7, \dots \), which are odd numbers with a linear progression.The total sum of the series after 80 terms is obtained by adding these two component series.Consequently, the sum of the series is \( 328160 \).
