Find the sum.
$1+1.13+1.13^2+1.13^3+\cdots+1.13^{12}$
The sum is $\square$ .
(Round to four decimal places as needed.)
Answer (2)
The sum of the series 1 + 1.13 + 1.1 3 2 + ⋯ + 1.1 3 12 is approximately 29.9847 . This is calculated using the formula for the sum of a geometric series. The key parameters involved are the first term, common ratio, and number of terms in the series.
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Identify the first term a = 1 , common ratio r = 1.13 , and number of terms n = 13 .
Apply the formula for the sum of a geometric series: S n = r − 1 a ( r n − 1 ) .
Substitute the values into the formula: S 13 = 1.13 − 1 1 ( 1.1 3 13 − 1 ) .
Calculate the sum and round to four decimal places: S 13 ≈ 29.9847 .
The sum is 29.9847 .
Explanation
Identifying the Series We are asked to find the sum of the series 1 + 1.13 + 1.1 3 2 + 1.1 3 3 + ⋯ + 1.1 3 12 . This is a geometric series. Let's identify the parameters of the geometric series.
Finding the Parameters The first term of the series is a = 1 . The common ratio is r = 1.13 . The number of terms in the series is n = 13 .
Stating the Formula The sum of a geometric series is given by the formula: S n = r − 1 a ( r n − 1 ) where S n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
Calculating the Sum Substituting the values a = 1 , r = 1.13 , and n = 13 into the formula, we get: S 13 = 1.13 − 1 1 ( 1.1 3 13 − 1 ) = 0.13 1.1 3 13 − 1 Now, we calculate 1.1 3 13 .
1.1 3 13 ≈ 4.89791076 So, S 13 = 0.13 4.89791076 − 1 = 0.13 3.89791076 ≈ 29.9839289 Rounding to four decimal places, we get 29.9839 .
Final Answer Therefore, the sum of the geometric series is approximately 29.9839 .
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 a fixed amount of money into an account each year, and the account earns a fixed interest rate. The total amount you will have at the end of a certain period can be calculated using the formula for the sum of a geometric series. This helps in financial planning and investment analysis.
