Find the sum of the following geometric series: [tex]$\sqrt{2}+\frac{1}{\sqrt{2}}+\frac{1}{2 \sqrt{2}}+\ldots \text { to } 8 \text { term }$[/tex]
Answer (2)
The sum of the given geometric series to 8 terms, starting with 2 and having a common ratio of 2 1 , is 128 255 2 .
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Identify the first term a = 2 and the common ratio r = 2 1 .
Verify that ∣ r ∣ < 1 , confirming the series converges.
Apply the formula for the sum of the first n terms: S n = a 1 − r 1 − r n .
Substitute n = 8 , a = 2 , and r = 2 1 to find the sum: S 8 = 128 255 2 .
The sum of the first 8 terms is 128 255 2 .
Explanation
Identifying the Geometric Series We are given a geometric series and asked to find the sum of the first 8 terms. Let's first identify the key components of the series.
Finding the Common Ratio The first term of the series, denoted as a , is 2 . The common ratio, denoted as r , is the ratio between consecutive terms. We can find it by dividing the second term by the first term: r = 2 2 1 = 2 1 ⋅ 2 1 = 2 1
Verifying Convergence and Stating the Sum Formula Since the absolute value of the common ratio is less than 1 (i.e., ∣ r ∣ = ∣ 2 1 ∣ < 1 ), the geometric series converges, and 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
Substituting Values into the Formula We are asked to find the sum of the first 8 terms, so n = 8 . Substituting the values a = 2 , r = 2 1 , and n = 8 into the formula, we get: S 8 = 2 1 − 2 1 1 − ( 2 1 ) 8 Now, let's simplify the expression.
Simplifying the Expression First, we calculate ( 2 1 ) 8 = 2 8 1 = 256 1 . Then, we have: S 8 = 2 1 − 2 1 1 − 256 1 = 2 2 1 256 256 − 1 = 2 2 1 256 255 = 2 ⋅ 256 255 ⋅ 2 = 2 ⋅ 128 255
Final Answer Therefore, the sum of the first 8 terms of the geometric series is 128 255 2 .
Examples
Geometric series are incredibly useful in many areas of science and engineering. For example, imagine a ball is dropped from a height of 1 meter, and each time it bounces, it reaches half of its previous height. The total distance the ball travels, both downwards and upwards, can be modeled as a geometric series. Understanding geometric series helps engineers design systems where energy loss occurs predictably, like in shock absorbers or electrical circuits.
