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Part 1
Determine whether the following series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms.
500plus100plus20plus4plustimestimestimes​; nequals200
Part 1
Determine whether the following series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms.
500plus100plus20plus4plustimestimestimes​; nequals200
Answers
GPT-4o mini
To determine whether the series \( 500 + 100 + 20 + 4 + \ldots \) is arithmetic or geometric, we need to define each type:
- An **arithmetic series** has a constant difference between consecutive terms. In other words, for a series to be arithmetic, \( a_{n+1} - a_n = d \) for all \( n \), where \( d \) is the common difference.
- A **geometric series** has a constant ratio between consecutive terms. For a series to be geometric, \( \frac{a_{n+1}}{a_n} = r \) for all \( n \), where \( r \) is the common ratio.
Now let's analyze the given series:
1. **Finding the ratios:**
- From the first term to the second term:
\[
\frac{100}{500} = \frac{1}{5}
\]
- From the second term to the third term:
\[
\frac{20}{100} = \frac{1}{5}
\]
- From the third term to the fourth term:
\[
\frac{4}{20} = \frac{1}{5}
\]
Since the ratio is consistently \( \frac{1}{5} \), this series is a geometric series.
2. **Identifying the first term and common ratio:**
- The first term (\( a \)) is \( 500 \).
- The common ratio (\( r \)) is \( \frac{1}{5} \).
3. **Evaluating the finite geometric series:**
The formula for the sum \( S_n \) of the first \( n \) terms of a geometric series is:
\[
S_n = a \frac{1 - r^n}{1 - r}
\]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
Now calculate \( S_{200} \):
- Here, \( a = 500 \), \( r = \frac{1}{5} \), and \( n = 200 \).
First, calculate \( r^n \):
\[
r^{200} = \left(\frac{1}{5}\right)^{200}
\]
This is a very small number approaching zero, so for practical purposes in calculating \( S_{200} \):
\[
S_{200} = 500 \frac{1 - \left(\frac{1}{5}\right)^{200}}{1 - \frac{1}{5}} = 500 \frac{1 - (very small number)}{\frac{4}{5}}
\]
The formula simplifies to:
\[
S_{200} \approx 500 \times \frac{5}{4} = 625
\]
So, the sum of the first 200 terms of the series is approximately:
\[
S_{200} \approx 625
\]
Therefore, the series is **geometric**, and the sum of the first 200 terms is approximately **625**.
- An **arithmetic series** has a constant difference between consecutive terms. In other words, for a series to be arithmetic, \( a_{n+1} - a_n = d \) for all \( n \), where \( d \) is the common difference.
- A **geometric series** has a constant ratio between consecutive terms. For a series to be geometric, \( \frac{a_{n+1}}{a_n} = r \) for all \( n \), where \( r \) is the common ratio.
Now let's analyze the given series:
1. **Finding the ratios:**
- From the first term to the second term:
\[
\frac{100}{500} = \frac{1}{5}
\]
- From the second term to the third term:
\[
\frac{20}{100} = \frac{1}{5}
\]
- From the third term to the fourth term:
\[
\frac{4}{20} = \frac{1}{5}
\]
Since the ratio is consistently \( \frac{1}{5} \), this series is a geometric series.
2. **Identifying the first term and common ratio:**
- The first term (\( a \)) is \( 500 \).
- The common ratio (\( r \)) is \( \frac{1}{5} \).
3. **Evaluating the finite geometric series:**
The formula for the sum \( S_n \) of the first \( n \) terms of a geometric series is:
\[
S_n = a \frac{1 - r^n}{1 - r}
\]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
Now calculate \( S_{200} \):
- Here, \( a = 500 \), \( r = \frac{1}{5} \), and \( n = 200 \).
First, calculate \( r^n \):
\[
r^{200} = \left(\frac{1}{5}\right)^{200}
\]
This is a very small number approaching zero, so for practical purposes in calculating \( S_{200} \):
\[
S_{200} = 500 \frac{1 - \left(\frac{1}{5}\right)^{200}}{1 - \frac{1}{5}} = 500 \frac{1 - (very small number)}{\frac{4}{5}}
\]
The formula simplifies to:
\[
S_{200} \approx 500 \times \frac{5}{4} = 625
\]
So, the sum of the first 200 terms of the series is approximately:
\[
S_{200} \approx 625
\]
Therefore, the series is **geometric**, and the sum of the first 200 terms is approximately **625**.