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!