Evaluate:
[tex]\begin{array}{r}
\sum_{n=1}^{10} 5(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 = 5 , the common ratio r = 3 , and the number of terms n = 10 .
Substitute these values into the formula for the sum of a geometric series: S = 1 − r a ( 1 − r n ) .
Calculate 3 10 = 59049 .
Calculate the sum: S = 1 − 3 5 ( 1 − 59049 ) = 147620 . The final answer is 147620 .
Explanation
Understanding the Problem We are asked to evaluate the sum ∑ n = 1 10 5 ( 3 ) n − 1 . This is a geometric series, and we can use the formula for the sum of a geometric series to find the answer.
Recalling the Formula The formula for the sum of a geometric series is given as 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 the Parameters In our series, the first term is a = 5 ( 3 ) 1 − 1 = 5 ( 3 ) 0 = 5 ( 1 ) = 5 . The common ratio is r = 3 . The number of terms is n = 10 .
Substituting the Values Now we substitute the values of a , r , and n into the formula for the sum of a geometric series: S = 1 − 3 5 ( 1 − 3 10 ) .
Calculating 3^10 We need to calculate 3 10 . The result is 3 10 = 59049 .
Calculating the Sum Now we can calculate the sum: S = 1 − 3 5 ( 1 − 59049 ) = − 2 5 ( − 59048 ) = − 2 − 295240 = 147620 .
Examples
Geometric series are not just abstract math; they appear in many real-world situations. For example, imagine you deposit $100 into a savings account each month that earns 5% interest compounded monthly. The total amount you have after a certain number of months can be calculated using the formula for the sum of a geometric series. Another example is calculating the total distance a bouncing ball travels before coming to rest, where each bounce is a fraction of the previous one. Understanding geometric series helps in predicting growth, decay, and accumulation in various scenarios.
The sum S of the geometric series is evaluated to be 147620 . This was done by identifying the parameters, using the geometric series sum formula, calculating the necessary powers, and substituting accordingly. Finally, the correct total is obtained through the formula calculations.
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