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Chapter 9: Problem 12

Evaluate the following geometric sums. $$\sum_{k=1}^{5}(-2.5)^{k}$$

Short Answer

Expert verified
Answer: The sum of the geometric sequence is \(-70.46875\).

Step by step solution

01

Write the geometric sum formula

To evaluate the geometric sum, we will use the formula: $$S_n = a_1\frac{1 - r^n}{1 - r}$$ Where \(S_n\) is the sum of the geometric sequence, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
02

Identify values of \(a_1\), \(r\), and \(n\)

In our given sum, we have: $$\sum_{k=1}^{5}(-2.5)^{k}$$ The first term is \(a_1 = -2.5^1 = -2.5\), the common ratio is \(r = -2.5\), and the number of terms is \(n=5\).
03

Substitute values into the geometric sum formula

Now, we will substitute the values of \(a_1\), \(r\), and \(n\) into the sum formula: $$S_n = -2.5\frac{1 - (-2.5)^5}{1 - (-2.5)}$$
04

Simplify the expression

First, we have to find the value of \((-2.5)^5\). Since it is an odd power, the result will be negative, and we have: $$(-2.5)^5 = -(2.5)^5 = -97.65625$$ Now, we can simplify the expression: $$S_n = -2.5\frac{1 - (-97.65625)}{1 - (-2.5)}$$ $$S_n = -2.5\frac{1 + 97.65625}{3.5}$$ $$S_n = -2.5\frac{98.65625}{3.5}$$
05

Calculate final sum

Finally, divide the numerator and the denominator and multiply the result by -2.5 to find the sum \(S_n\): $$S_n = -2.5\left(\frac{98.65625}{3.5}\right) = -2.5(28.1875)$$ $$S_n = -70.46875$$ So, the sum of the given geometric sequence is \(S_5 = -70.46875\).

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Key Concepts

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

Geometric Sum Formula
In mathematics, geometric sequences have a neat formula to evaluate the sum of their terms. This special formula comes in handy, especially when calculating sums with a finite number of terms. The geometric sum formula is given by:\[ S_n = a_1\frac{1-r^n}{1-r} \]where:
  • \( S_n \) represents the sum of the geometric sequence.
  • \( a_1 \) is the first term of the sequence.
  • \( r \) is the common ratio between successive terms.
  • \( n \) indicates the number of terms in the sequence.
This formula becomes very powerful as it allows us to evaluate sums without having to add each term manually. Simply plug in the values for \( a_1 \), \( r \), and \( n \) to find the sum swiftly.
Common Ratio
The common ratio is a fundamental part of understanding geometric sequences. It is a consistent multiplier between each term of the sequence. Let's elaborate on it using our exercise's example.Take the sequence defined by the summation \( \sum_{k=1}^{5}(-2.5)^{k} \). Here, each successive term is calculated by multiplying the preceding term by \(-2.5\). Hence, the common ratio \( r \) is \(-2.5\).
  • The common ratio helps determine how quickly the sequence grows or shrinks.
  • If \(|r| < 1\), the sequence terms decrease in magnitude as you progress.
  • If \(|r| > 1\), the sequence terms expand significantly.
Understanding the nature of the common ratio can give insights into the behavior of the whole sequence.
Sequence Terms
Geometric sequences comprise terms that follow a pattern established by a constant ratio. Each term can be computed by multiplying the previous term by the common ratio. Considering our problem, the sequence turns out as:
  • \(-2.5\)
  • \((-2.5)^2 = 6.25\)
  • \((-2.5)^3 = -15.625\)
  • \((-2.5)^4 = 39.0625\)
  • \((-2.5)^5 = -97.65625\)
The sequence is alternating in signs because of the negative common ratio \(-2.5\). Each term's absolute value grows because the magnitude of the common ratio is greater than one. Recognizing and listing the terms is crucial when tackling problems involving sequences and series.
Evaluating Sums
When evaluating the sum of a geometric sequence, it's essential to correctly apply the formula with identified values. In our example, the geometric sum formula is used with the identified first term \( a_1 = -2.5 \), common ratio \( r = -2.5 \), and number of terms \( n = 5 \).Substituting these into the equation:\[ S_n = -2.5\frac{1 - (-2.5)^5}{1 - (-2.5)} \]We performed the calculations:
  • Calculate the power: \((-2.5)^5 = -97.65625\)
  • Simplify the expression: \[ S_n = -2.5\frac{98.65625}{3.5} \]
  • Compute the fraction: \( \frac{98.65625}{3.5} = 28.1875\)
  • Find the final sum: \( S_n = -2.5 \times 28.1875 = -70.46875 \)
Hence, the sum of the terms in this sequence is \( S_5 = -70.46875 \). Carefully following each calculation step ensures accuracy in evaluating geometric sums.

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Most popular questions from this chapter

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=0}^{\infty} x^{k}$$

The expression where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+1 / a_{n},\) for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim a_{n}\). b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of where \(a\) and \(b\) are positive real numbers.

Evaluate the limit of the following sequences. $$a_{n}=\cos \left(0.99^{n}\right)+\frac{7^{n}+9^{n}}{63^{n}}$$

Consider the expression \(\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}},\) where the process continues indefinitely. a. Show that this expression can be built in steps using. the recurrence relation \(a_{0}=1, a_{n+1}=\sqrt{1+a_{n}},\) for \(n=0,1,2,3, \ldots .\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n}\). b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Estimate the limit of the sequence. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. e. Repeat the preceding analysis for the expression \(\sqrt{p+\sqrt{p+\sqrt{p+\sqrt{p+\cdots}}}}\) where \(p > 0 .\) Make a table showing the approximate value of this expression for various values of \(p .\) Does the expression seem to have a limit for all positive values of \(p ?\)

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=4 a_{n}\left(1-a_{n}\right) ; a_{0}=0.5, n=0,1,2, \dots$$
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