The problem asks to find the sum $S_{15}$ for the arithmetic series given by:

$5 + 9 + 13 + 17 + \dots$

This is an arithmetic series where:

• The first term $a_1 = 5$
• The common difference $d = 4$ (since $9 - 5 = 4$)

The sum of the first $n$ terms of an arithmetic series is given by the formula:

$S_n = \frac{n}{2} \times \left(2a_1 + (n-1) \times d\right)$

Here, we are asked to find $S_{15}$, so $n = 15$. Plugging in the values:

$S_{15} = \frac{15}{2} \times \left(2 \times 5 + (15-1) \times 4\right)$

$S_{15} = \frac{15}{2} \times \left(10 + 14 \times 4\right)$

$S_{15} = \frac{15}{2} \times \left(10 + 56\right)$

$S_{15} = \frac{15}{2} \times 66$

$S_{15} = \frac{990}{2} = 495$

So, the sum $S_{15}$ is 495.

Would you like more details or have any other questions?

Here are 5 related questions:

1. How do you derive the sum formula for an arithmetic series?
2. What happens if the common difference $d$ is negative in an arithmetic series?
3. Can you determine the 15th term $a_{15}$ of this series?
4. How does the formula for the sum of a series change if we only know the first and last terms?
5. What is the sum $S_n$ of the series if $n = 20$?

Tip: Always verify the common difference $d$ in arithmetic series problems to avoid errors in calculation.