The expression given in the image is:

$3 + 7 + 11 + 15 + \ldots + 99$

This sequence is an arithmetic series where:

• The first term $a = 3$
• The common difference $d = 4$
• The last term $l = 99$

To find the sum of this series, we can use the formula for the sum of an arithmetic series:

$S_n = \frac{n}{2} \times (a + l)$

First, we need to find the number of terms, $n$, in the series. The formula for the $n$-th term of an arithmetic sequence is:

$a_n = a + (n - 1) \cdot d$

Setting $a_n = 99$:

$99 = 3 + (n - 1) \cdot 4$ $96 = (n - 1) \cdot 4$ $n - 1 = 24$ $n = 25$

There are 25 terms in the series.

Now, substitute into the sum formula:

$S_{25} = \frac{25}{2} \times (3 + 99)$ $S_{25} = \frac{25}{2} \times 102$ $S_{25} = 25 \times 51$ $S_{25} = 1275$

Thus, the sum of the series is:

$\boxed{1275}$

Would you like further details on this solution or have any questions?

Here are five related questions to deepen understanding:

1. How would the sum change if the series ended at a different term?
2. What if the common difference was different? How would that affect the result?
3. Can you generalize the formula for any arithmetic sequence sum?
4. How can this method be applied to find the sum of only the even or odd terms of a similar sequence?
5. What would happen if the sequence was geometric instead of arithmetic?

Tip: In an arithmetic series, always find the number of terms first if they aren’t given.