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Chapter 10: Problem 63

Evaluate the sum. For each sum, state whether it is arithmetic or geometric. Depending on your answer, state the value of d or \(r\). $$\sum_{k=1}^{5} 2(0.5)^{k}$$

Short Answer

Expert verified
The given series is a geometric series with a common ratio of \(r = 0.5\). After performing the calculations, the sum of this geometric series can be obtained.

Step by step solution

01

Identify type of series

Look at the given sum, \(\sum_{k=1}^{5} 2(0.5)^{k}\). Notice how each term is half of the preceding term which indicates that this is a geometric series.
02

Compute common ratio, r

The common ratio of a geometric series is the factor between consecutive terms. Here, the value inside the parenthesis raised to the power of k is the ratio. Thus, \(r = 0.5\).
03

Apply formula for the sum of a finite geometric series

We can use this formula to compute the sum of a finite geometric series: \(S_n = a(r^n - 1) / (r - 1)\), where \(S_n\) is the sum of the first n terms, a is the first term in the sequence, r is the common ratio, and n is the number of terms. Here, \(a = 2*0.5 = 1\), \(r = 0.5\), and \(n = 5\). Substituting these values into the formula gives us: \(S_5 = 1(0.5^5 - 1) / (0.5 - 1)\).
04

Calculate the sum

Carry out the calculations from the previous step. This will give us the sum of the finite geometric series.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
In a geometric series, each term is obtained by multiplying the previous term by a fixed and consistent value known as the common ratio. This ratio is crucial in defining the nature of the geometric series. For example, in the series given by the problem,
  • First term: 2(0.5)
  • Second term: 2(0.5)^2
  • Third term: 2(0.5)^3
Here, the common ratio is 0.5, as each term is half of the preceding one. The identification of a common ratio allows us to determine the subsequent terms easily. It is a constant (much like "d" in an arithmetic series), and it tells us the rate of progression between terms. Understanding the common ratio helps in predicting future terms and is foundational for calculating the sum of the series.
Finite Geometric Series
A finite geometric series is a collection of terms in a sequence where there are a specific number of terms, and each successive term is the product of the previous term and a constant known as the common ratio. Unlike an infinite series, a finite geometric series has an end point, making calculations more straightforward.
A finite geometric series typically looks like:
  • First term: \(a\)
  • Second term: \(ar\)
  • Third term: \(ar^2\)
  • ... up to the nth term \(ar^{n-1}\)
This form highlights how each term relates proportionally to others through multiplication by the common ratio \(r\). In this example, with \(n = 5\), we have five terms starting with \(2(0.5)\). Recognizing the parameters of finite geometric series helps us efficiently apply formulas to ascertain the total sum.
Sum of Series
The sum of a finite geometric series can be calculated using a specialized formula, which is crucial when the series involves multiple terms. The formula is given by:\[ S_n = \frac{a(r^n - 1)}{r - 1} \]where:
  • \(S_n\) denotes the sum of the first \(n\) terms.
  • \(a\) represents the first term.
  • \(r\) is the common ratio.
  • \(n\) is the number of terms.
In the problem's context, the first term \(a\) can be calculated as \(2 \times 0.5 = 1\), \(r\) is 0.5, and \(n\) is 5. Plugging these into the formula allows us to solve for the sum efficiently:\[ S_5 = \frac{1(0.5^5 - 1)}{0.5 - 1} \]Using this method aligns perfectly with understanding how geometric series are built, helping in visualizing and solving summation questions with precision.

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