using the formula and manual method. Find the sum of 5 terms of a geometric sequence with the first term 83 and last term 1

To find the sum of the first 5 terms of a geometric sequence where the first term \( a = 83 \) and the last term \( l = 1 \), we need to follow a structured approach.

### 1. General Formula for Sum of Terms in a Geometric Sequence
The sum \( S_n \) of the first \( n \) terms of a geometric sequence is given by:

\[
S_n = a \frac{1 - r^n}{1 - r} \quad \text{(if \( r \neq 1 \))} 
\]
Where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

### 2. Step-by-Step Method to Solve the Problem

#### Step 1: Identify Key Information
- First term \( a = 83 \),
- Last term \( l = 1 \),
- Number of terms \( n = 5 \).

#### Step 2: Find the Common Ratio \( r \)
Since we know the first and last terms, we can use the formula for the \( n \)-th term of a geometric sequence:

\[
l = a \cdot r^{n-1}
\]

Substituting the given values:

\[
1 = 83 \cdot r^{5-1} \quad \Rightarrow \quad 1 = 83 \cdot r^4
\]

Solving for \( r \):

\[
r^4 = \frac{1}{83} \quad \Rightarrow \quad r = \left( \frac{1}{83} \right)^{\frac{1}{4}}
\]

Now, calculate the value of \( r \).

#### Step 3: Calculate the Sum of the First 5 Terms
Once \( r \) is found, we can substitute it into the sum formula for the first 5 terms:

\[
S_5 = 83 \cdot \frac{1 - r^5}{1 - r}
\]

Let me calculate \( r \) and \( S_5 \).The common ratio \( r \) of the geometric sequence is approximately \( 0.3313 \), and the sum of the first 5 terms of the sequence is approximately \( 123.63 \).

### Summary:
- Common ratio \( r \approx 0.3313 \),
- Sum of the first 5 terms \( S_5 \approx 123.63 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How is the common ratio \( r \) derived in geometric sequences?
2. What happens if the common ratio \( r \) is negative?
3. How can we generalize the sum formula for any number of terms?
4. What is the sum if the sequence extends beyond 5 terms?
5. How does the sum of an infinite geometric sequence work when \( |r| < 1 \)?

**Tip:** In geometric sequences, the value of \( r \) greatly affects whether the terms increase, decrease, or alternate between positive and negative values!

Discover how to find the sum of the first 5 terms of a geometric sequence with a first term of 83 and a last term of 1. This problem involves calculating the common ratio and applying the geometric series sum formula. Ideal for students in Grades 9-12.

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