Find the value of the series
\[1 + 4 + 2 + 8 + 3 + 12 + 4 + 16 + \cdots + 24 + 96 + 25 + 100.\]
The series can be split into two parts:
1 + 4 + 2 + 8 + 3 + 12 + ... + 24 + 96 25 + 100
The first part is an arithmetic series with first term 1, common difference 2, and 25 terms. The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum of the first part is
(1 + 96)/2 * 25 = 2475
The second part is a geometric series with first term 25, common ratio 4, and 2 terms. The sum of a geometric series is equal to the first term divided by 1 minus the common ratio, so the sum of the second part is
25 / (1 - 4) = 100
The sum of the entire series is equal to the sum of the first part plus the sum of the second part, or 2475 + 100 = 2575.
