Find the sum of the first 6 terms of the following sequence. Round to the nearest hundredth if necessary. 4 12, 4, Sum of a finite geometric series: Sn αι – αγγ 1-r =
Find the sum of the first 6 terms of the following sequence. Round to the nearest hundredth if necessary. 4 12, 4, Sum of a finite geometric series: Sn αι – αγγ 1-r =
Chapter12: Sequences, Series And Binomial Theorem
Section: Chapter Questions
Problem 328PT: Find the first term and common difference of an arithmetic sequence whose ninth term is -1 and the...
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![### Geometric Sequence Summation
**Problem Statement:**
Find the sum of the first 6 terms of the following sequence. Round to the nearest hundredth if necessary.
Sequence:
\[ 12, \ 4, \ \frac{4}{3}, \ \ldots \]
**Formula for the Sum of a Finite Geometric Series:**
\[ S_n = \frac{a_1 (1 - r^n)}{1 - r} \]
Where:
- \( S_n \) = Sum of the first \( n \) terms
- \( a_1 \) = First term of the series
- \( r \) = Common ratio between the terms
- \( n \) = Number of terms to be summed
### Explanation:
In a geometric series, each term after the first is the product of the previous term and a constant known as the common ratio (\( r \)).
This formula helps in efficiently calculating the sum of the first \( n \) terms of the sequence without the need to manually add each term. The sum is especially useful for sequences with a large number of terms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78f36c12-bb42-44cf-b4d7-4a7cf773a344%2Fc47bc97a-f06f-43b5-81ed-ebe5f6d00be1%2Fclgtu8k_processed.png&w=3840&q=75)
Transcribed Image Text:### Geometric Sequence Summation
**Problem Statement:**
Find the sum of the first 6 terms of the following sequence. Round to the nearest hundredth if necessary.
Sequence:
\[ 12, \ 4, \ \frac{4}{3}, \ \ldots \]
**Formula for the Sum of a Finite Geometric Series:**
\[ S_n = \frac{a_1 (1 - r^n)}{1 - r} \]
Where:
- \( S_n \) = Sum of the first \( n \) terms
- \( a_1 \) = First term of the series
- \( r \) = Common ratio between the terms
- \( n \) = Number of terms to be summed
### Explanation:
In a geometric series, each term after the first is the product of the previous term and a constant known as the common ratio (\( r \)).
This formula helps in efficiently calculating the sum of the first \( n \) terms of the sequence without the need to manually add each term. The sum is especially useful for sequences with a large number of terms.
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