Find the sum of the following geometric series: [tex]$\sqrt{2}+\frac{2}{\sqrt{2}}+\frac{1}{2 \sqrt{2}}+\ldots \text { to } 8 \text { term }$[/tex]
Answer (2)
The sum of the geometric series 2 + 2 1 + 2 2 1 + … for the first 8 terms is 128 255 2 . This was calculated using the geometric series sum formula after identifying the first term and common ratio. The result shows how the terms relate through their common ratio, allowing for the use of the summation formula.
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Identify the first term a = 2 and the common ratio r = 2 1 .
Use the formula for the sum of a geometric series: S n = a 1 − r 1 − r n .
Substitute a = 2 , r = 2 1 , and n = 8 into the formula: S 8 = 2 1 − 2 1 1 − ( 2 1 ) 8 .
Simplify the expression to find the sum: S 8 = 128 255 2 .
128 255 2
Explanation
Problem Analysis We are given a geometric series and asked to find the sum of the first 8 terms. The series is 2 + 2 2 + 2 2 1 + … . To find the sum of a geometric series, we need to identify the first term, a , and the common ratio, r . Then we can use the formula for the sum of the first n terms of a geometric series: S n = a 1 − r 1 − r n .
Finding the Common Ratio The first term is a = 2 . To find the common ratio, we divide the second term by the first term: r = 2 2 2 = 2 2 \t ⋅ 2 1 = 2 2 = 1. However, if we divide the third term by the second term, we get r = 2 2 2 2 1 = 2 2 1 \t ⋅ 2 2 = 4 1 . Since the ratio between consecutive terms is not constant, there must be a typo in the problem. Let's assume the series is 2 + 2 1 + 2 2 1 + … . Then the common ratio is r = 2 2 1 = 2 1 \t ⋅ 2 1 = 2 1 .
Calculating the Sum Now we have a = 2 and r = 2 1 . We want to find the sum of the first 8 terms, so n = 8 . Plugging these values into the formula for the sum of a geometric series, we get S 8 = 2 1 − 2 1 1 − ( 2 1 ) 8 = 2 2 1 1 − 256 1 = 2 2 ( 1 − 256 1 ) = 2 2 ( 256 255 ) = 128 255 2 .
Final Answer Therefore, the sum of the first 8 terms of the geometric series is 128 255 2 .
Examples
Geometric series are useful in many areas of mathematics and have practical applications in physics, engineering, biology, economics, computer science, and finance. 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 this annuity 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 traveled by a bouncing ball, where each bounce is a fraction of the previous bounce's height.
