Evaluate:
[tex]\begin{array}{c}
\sum_{n=1}^6 4(3)^{n-1} \
S=[?]\end{array}[/tex]
Remember: for a geometric series, [tex]S=\frac{a\left(1-r^n\right)}{1-r}[/tex]
Answer (2)
Identify the first term a = 4 , the common ratio r = 3 , and the number of terms n = 6 .
Substitute the values into the formula for the sum of a geometric series: S = 1 − r a ( 1 − r n ) .
Calculate the sum: S = 1 − 3 4 ( 1 − 3 6 ) = 1456 .
The sum of the geometric series is 1456 .
Explanation
Understanding the problem We are asked to evaluate the sum of a geometric series: n = 1 ∑ 6 4 ( 3 ) n − 1
Recalling the formula The general formula for the sum of a geometric series is given by: S = 1 − r a ( 1 − r n ) where:
a is the first term of the series,
r is the common ratio,
n is the number of terms.
Identifying the parameters In our case, we have:
a = 4 ( 3 ) 1 − 1 = 4 ( 3 ) 0 = 4
r = 3
n = 6
Calculating the sum Now, we substitute these values into the formula: S = 1 − 3 4 ( 1 − 3 6 ) S = − 2 4 ( 1 − 729 ) S = − 2 4 ( − 728 ) S = 2 ( 728 ) S = 1456
Final Answer Therefore, the sum of the geometric series is 1456.
Examples
Geometric series can be used to model various real-world phenomena, such as the growth of a population, the decay of a radioactive substance, or the calculation of compound interest. For instance, if you invest $1000 in an account that pays 5% interest compounded annually, the amount of money you have each year forms a geometric sequence. Understanding geometric series allows you to predict the future value of your investment or the total amount of money you will earn over a certain period.
The sum of the series ∑ n = 1 6 4 ( 3 ) n − 1 can be calculated using the formula for a geometric series, yielding a final answer of 1456 . This is obtained by identifying the first term, common ratio, and number of terms, then substituting these values into the formula. Thus, the answer is 1456 .
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