The sequence you provided is $4, 4, 4, 4, \dots$, which is a constant sequence. This means it's not a geometric progression, but a constant arithmetic sequence. However, the formula you mentioned:

$S_n = A_1 \frac{1 - r^n}{1 - r}$

is for the sum of a geometric series, which applies when the common ratio $r \neq 1$. For this constant sequence, the ratio $r = 1$, and the sum formula would not apply directly.

Since each term in the sequence is $4$, we can instead calculate the sum using a simple formula for the sum of an arithmetic sequence where each term is the same:

$S_n = n \times A_1$

Where:

• $n = 50$ (the number of terms),
• $A_1 = 4$ (the value of each term).

Thus, the sum of the first 50 terms is:

$S_{50} = 50 \times 4 = 200$

Final Answer:

The sum of the first 50 terms is $200$.

Would you like further details or explanations?

Here are 5 related questions to deepen your understanding:

1. How would the formula for a geometric series apply if the ratio $r \neq 1$?
2. What is the formula for the sum of the first $n$ terms of an arithmetic sequence?
3. How does the sum of an infinite geometric series differ from a finite one?
4. What happens to the sum of terms when the common ratio $r > 1$ in a geometric series?
5. How can you determine if a sequence is arithmetic or geometric based on its terms?

Tip: When working with constant sequences, remember that the sum is simply the value of the term multiplied by the number of terms!