The problem asks for the sum of the series 25 + 26 + ... + 75. This is an arithmetic progression.

Identifying Series Parameters

• First term, $a = 25$.
• Last term, $l = 75$.
• Common difference, $d = 1$ (since each term increases by 1).

Calculating the Number of Terms (n)

Use the formula for the $n$-th term of an arithmetic progression: $l = a + (n-1)d$.

Substitute the known values:

$75 = 25 + (n-1) \times 1$

Solve for $n$:

$75 - 25 = n - 1$

$50 = n - 1$

$n = 50 + 1 = 51$

There are 51 terms in the series.

Calculating the Sum of the Series (S)

Use the formula for the sum of an arithmetic progression: $S_n = \frac{n}{2}(a + l)$.

Substitute the values of $n$, $a$, and $l$:

$S_{51} = \frac{51}{2}(25 + 75)$

$S_{51} = \frac{51}{2}(100)$

$S_{51} = 51 \times 50$

$S_{51} = 2550$

The sum of the series is 2550.