Evaluate:
[tex]\begin{array}{c}
\sum_{n=1}^4 3(7)^{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 , the common ratio r = 7 , and the number of terms n = 4 .
Apply the formula for the sum of a geometric series: S = 1 − r a ( 1 − r n ) .
Substitute the values into the formula: S = 1 − 7 3 ( 1 − 7 4 ) .
Calculate the sum: S = 1200 . The final answer is 1200 .
Explanation
Understanding the Problem We are asked to evaluate the sum of a geometric series: n = 1 ∑ 4 3 ( 7 ) n − 1
Identifying Parameters We can recognize this as a geometric series with first term a = 3 ( 7 ) 1 − 1 = 3 ( 7 ) 0 = 3 , common ratio r = 7 , and number of terms n = 4 .
Stating the Formula The formula for the sum of a geometric series is given by: S = 1 − r a ( 1 − r n ) where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Substituting Values Substituting the values a = 3 , r = 7 , and n = 4 into the formula, we get: S = 1 − 7 3 ( 1 − 7 4 ) We calculate 7 4 = 2401 .
Calculating the Sum Substituting this value back into the formula, we have: S = 1 − 7 3 ( 1 − 2401 ) = − 6 3 ( − 2400 ) S = − 6 − 7200 = 1200 Thus, the sum of the series is 1200.
Final Answer Therefore, the value of the sum is: S = 1200
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 annually, compounded monthly. The total amount you'll have after a certain number of months can be calculated using the formula for the sum of a geometric series. Or consider the spread of a virus, where each infected person infects a certain number of new people. The total number of infected people can be modeled using a geometric series. Understanding geometric series helps us predict and analyze these kinds of phenomena.
The sum of the geometric series S = ∑ n = 1 4 3 ( 7 ) n − 1 evaluates to 1200. This is calculated using the geometric series sum formula, where the first term is 3, the common ratio is 7, and the number of terms is 4. By substituting these values into the formula, we find that the total sum is indeed 1200.
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