Find the sum of the first 50 terms, (4,4,4,4,4,...) using formula Sn = A1 (1-r^n) / 1-r

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!

Learn how to find the sum of the first 50 terms of a constant sequence (4, 4, 4, ...) using the arithmetic sequence summation formula. This example also discusses when to use geometric series formulas and when they don't apply. Suitable for Grades 6-8.

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