Evaluate:
[tex]\begin{array}{c}
\sum_{n=1}^5 3(4)^{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 = 3 , common ratio r = 4 , and number of terms n = 5 .
Substitute these values into the formula for the sum of a geometric series: S = 1 − r a ( 1 − r n ) .
Calculate 4 5 = 1024 and substitute into the formula: S = 1 − 4 3 ( 1 − 1024 ) .
Simplify the expression to find the sum: S = 1023 , so the final answer is 1023 .
Explanation
Understanding the Problem We are asked to evaluate the sum of a geometric series: n = 1 ∑ 5 3 ( 4 ) n − 1 We are also given the formula for the sum of a geometric series: S = 1 − r a ( 1 − r n ) where a is the first term, r is the common ratio, and n is the number of terms.
Identifying Parameters First, we need to identify the first term a , the common ratio r , and the number of terms n in the given series.
The first term is when n = 1 :
a = 3 ( 4 ) 1 − 1 = 3 ( 4 ) 0 = 3 ( 1 ) = 3 The common ratio is the factor by which each term is multiplied to get the next term. In this case, it is r = 4 .
The number of terms is n = 5 .
Calculating the Sum Now, we substitute the values of a , r , and n into the formula for the sum of a geometric series: S = 1 − r a ( 1 − r n ) = 1 − 4 3 ( 1 − 4 5 ) We know that 4 5 = 1024 , so we can substitute this value into the equation: S = 1 − 4 3 ( 1 − 1024 ) = − 3 3 ( − 1023 ) Now, we simplify the expression: S = − 3 − 3069 = 1023
Final Answer Therefore, the sum of the geometric series is 1023. n = 1 ∑ 5 3 ( 4 ) n − 1 = 1023
Examples
Geometric series are useful in many areas of mathematics and have practical applications in finance, physics, and computer science. For example, calculating the future value of an annuity involves summing a geometric series. Suppose you deposit $100 each month into an account that earns 5% annual interest, compounded monthly. The future value of your deposits can be calculated using the formula for the sum of a geometric series, where each term represents the value of a deposit at a future point in time. Understanding geometric series helps in predicting the long-term growth of investments and in making informed financial decisions.
The sum of the series ∑ n = 1 5 3 ( 4 ) n − 1 is 1023. This was evaluated by identifying the first term, common ratio, and number of terms, then applying the formula for the sum of a geometric series. After simplifying, we conclude that the total sum is 1023.
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