Question
An expression for the Sum of a Finite Geometric Series is shown.
3−3(2)^5/1−2
Complete each sentence:
(4 points)
The first term in the sequence is:
The common ratio is:
There are
terms in the sum.
The sum of the finite series is:
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3−3(2)^5/1−2
Complete each sentence:
(4 points)
The first term in the sequence is:
The common ratio is:
There are
terms in the sum.
The sum of the finite series is:
Skip to navigation
Answers
GPT-4o mini
To analyze the given expression for the sum of a finite geometric series, we can refer to the formula for the sum of a finite geometric series:
\[
S_n = a \frac{1 - r^n}{1 - r}
\]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term in the series,
- \( r \) is the common ratio,
- \( n \) is the number of terms.
Let's break down your expression step-by-step:
Given:
\[
S_n = 3 - \frac{3(2^5)}{1 - 2}
\]
We can rewrite it:
\[
S_n = 3 - \frac{3(32)}{-1} = 3 + 96 = 99
\]
From the expression, we can deduce the following:
1. **The first term in the sequence is:** The first term \( a \) can be identified from the first part of the series. Looking at the expression we have \( 3 \). So:
**Answer:** The first term in the sequence is: **3**.
2. **The common ratio is:** The common ratio \( r \) can be determined from how we derive the series. In this expression, we see that \( 2 \) is the root of \( 2^5 \), which implies \( r=2 \).
**Answer:** The common ratio is: **2**.
3. **There are** \( n \) **terms in the sum:** The \( n \) can be identified from the exponent of \( r \) in the term \( 2^5 \). Thus, \( n=5 \).
**Answer:** There are **6** terms in the sum (including the first term which represents the series from \( n=0 \) to \( n=5\)).
4. **The sum of the finite series is:** From our calculation earlier, the sum of the finite series \( S_n \) comes out to be \( 99 \).
**Answer:** The sum of the finite series is: **99**.
To summarize:
- The first term in the sequence is: **3**.
- The common ratio is: **2**.
- There are **6** terms in the sum.
- The sum of the finite series is: **99**.
\[
S_n = a \frac{1 - r^n}{1 - r}
\]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term in the series,
- \( r \) is the common ratio,
- \( n \) is the number of terms.
Let's break down your expression step-by-step:
Given:
\[
S_n = 3 - \frac{3(2^5)}{1 - 2}
\]
We can rewrite it:
\[
S_n = 3 - \frac{3(32)}{-1} = 3 + 96 = 99
\]
From the expression, we can deduce the following:
1. **The first term in the sequence is:** The first term \( a \) can be identified from the first part of the series. Looking at the expression we have \( 3 \). So:
**Answer:** The first term in the sequence is: **3**.
2. **The common ratio is:** The common ratio \( r \) can be determined from how we derive the series. In this expression, we see that \( 2 \) is the root of \( 2^5 \), which implies \( r=2 \).
**Answer:** The common ratio is: **2**.
3. **There are** \( n \) **terms in the sum:** The \( n \) can be identified from the exponent of \( r \) in the term \( 2^5 \). Thus, \( n=5 \).
**Answer:** There are **6** terms in the sum (including the first term which represents the series from \( n=0 \) to \( n=5\)).
4. **The sum of the finite series is:** From our calculation earlier, the sum of the finite series \( S_n \) comes out to be \( 99 \).
**Answer:** The sum of the finite series is: **99**.
To summarize:
- The first term in the sequence is: **3**.
- The common ratio is: **2**.
- There are **6** terms in the sum.
- The sum of the finite series is: **99**.