The image you uploaded contains a mathematical problem asking to determine the sum of the series:

$\sum_{n=4}^{20} (4n + 3)$

To solve this, we first calculate the sum of the $4n$ terms and the sum of the $3$ terms separately.

1. Sum of $4n$ from $n=4$ to $n=20$: The sum of a sequence of multiples of a constant can be calculated by factoring out the constant and using the sum formula for consecutive integers, which is $\frac{n(n + 1)}{2}$ where $n$ is the number of terms.

   • Calculate the sum $S_n = 4 + 8 + 12 + \ldots + 80$.
   • This series is an arithmetic sequence where $a = 4n$, starting at $n=4$. To find the sum, we note: $S_n = 4 \cdot \left(\frac{(20-4+1)(4+80)}{2}\right)$

2. Sum of the constant term $3$, added 17 times (from $n=4$ to $n=20$): $S_{const} = 3 \times 17$

Adding these sums gives us the total sum of the series. Let's compute this.The sum of the series $\sum_{n=4}^{20} (4n + 3)$ is 867.

Would you like more details on any part of the solution, or do you have any other questions?

Here are some related questions you might consider:

1. How would the sum change if the series started from $n=1$?
2. What if the constant term were different, for example, $5$ instead of $3$?
3. Can you calculate the average value of the terms in this series?
4. What if the series were extended to $n=25$? How would that affect the sum?
5. How does changing the coefficient of $n$ (from 4 to another number) affect the sum of the series?

Tip: When solving series problems, separating variable terms from constants can simplify calculations, as seen with the terms $4n$ and $3$ in this problem.