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Chapter 10: Problem 15

Write the first five terms of each geometric series. $$a_{1}=\frac{2}{3} \quad r=\frac{1}{2}$$

Short Answer

Expert verified
The first five terms are: \( \frac{2}{3}, \frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24} \).

Step by step solution

01

Understand the Formula for a Geometric Series

In a geometric series, each term can be found by multiplying the previous term by a common ratio, denoted as \( r \). The general formula for the \( n \)-th term of a geometric series is: \[ a_{n} = a_{1} imes r^{n-1} \] where \( a_{1} \) is the first term, and \( n \) is the term number.
02

Calculate the First Term

The first term \( a_{1} \) is given as \( \frac{2}{3} \). Therefore, the first term of the series is \( \frac{2}{3} \).
03

Calculate the Second Term

To find the second term \( a_{2} \), use the formula:\[ a_{2} = a_{1} imes r = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} \]
04

Calculate the Third Term

Now, calculate the third term \( a_{3} \) using the formula:\[ a_{3} = a_{2} imes r = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \]
05

Calculate the Fourth Term

Next, find the fourth term \( a_{4} \) using:\[ a_{4} = a_{3} \times r = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \]
06

Calculate the Fifth Term

Finally, calculate the fifth term \( a_{5} \) using:\[ a_{5} = a_{4} \times r = \frac{1}{12} \times \frac{1}{2} = \frac{1}{24} \]

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Key Concepts

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

Common Ratio
A common ratio is a key element in understanding geometric sequences. It is the factor by which we multiply one term to get the next term in the sequence. The common ratio is constant throughout a geometric sequence, which makes geometric sequences predictable and easy to analyze. Let's understand more about the nature of the common ratio:
  • If the common ratio is greater than 1, the terms of the sequence will increase.
  • If the common ratio is between 0 and 1, as in this exercise where the common ratio is \( \frac{1}{2} \), the terms will decrease.
  • If the common ratio is negative, the terms will alternate in sign, creating a back-and-forth pattern.
Knowing the common ratio allows us to easily generate any term in the sequence by multiplying a term by this ratio repeatedly.
Geometric Sequence
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero value called the common ratio. In this exercise, we have:
  • First term \( a_1 = \frac{2}{3} \).
  • Common ratio \( r = \frac{1}{2} \).
To illustrate, the sequence starts with \( \frac{2}{3} \), then each subsequent term is half of the previous term. This gives us the terms: \( \frac{2}{3}, \frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24} \).Geometric sequences can be finite or infinite, depending on how many terms are considered. They are particularly useful for modeling situations where growth or decay occurs at a constant percentage rate.
Nth Term Formula
The nth term formula in a geometric sequence is a powerful tool that allows us to find any term in the sequence without listing all the previous terms. The formula is:\[ a_n = a_1 \times r^{n-1} \]Where:
  • \( a_n \) is the nth term.
  • \( a_1 \) is the first term of the sequence.
  • \( r \) is the common ratio.
  • \( n \) is the term number.
To use the formula, plug in the values of \( a_1 \), \( r \), and \( n \). For example, to find the third term \( a_3 \), the formula becomes:\[ a_3 = \frac{2}{3} \times \left(\frac{1}{2}\right)^{3-1} = \frac{2}{3} \times \frac{1}{4} = \frac{1}{6} \]This formula makes it simple to calculate any term, saving time and effort compared to multiplying terms manually.

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Most popular questions from this chapter

Upon graduation Sheldon decides to go to work for a local police department. His starting salary is 30,000 dollars per year, and he expects to get a \(3 \%\) raise per year. Write the recursion formula for a sequence that represents his annual salary after \(n\) years on the job. Assume \(n=0\) represents his first year making 30,000 dollars.

Let \(a_{n}=\sqrt{a_{n-1}}\) for \(n \geq 2\) and \(a_{1}=7 .\) Find the first five terms of this sequence and make a generalization for the \(n\) th term.

A player in the NFL typically has a career that lasts 3 years. The practice squad makes the league minimum of 275,000 dollars (2004) in the first year, with a 75,000 dollars raise per year. Write the general term of a sequence \(a_{n}\) that represents the salary of an NFL player making the league minimum during his entire career. Assuming \(n=1\) corresponds to the first year, what does \(\sum_{n=1}^{3} a_{n}\) represent?

In calculus, when estimating certain integrals, we use sums of the form \(\sum_{i=1}^{n} f\left(x_{i}\right) \Delta x,\) where \(f\) is a function and \(\Delta x\) is a constant. Find the indicated sum. $$\sum_{i=1}^{85} f\left(x_{i}\right) \Delta x, \text { where } f\left(x_{i}\right)=6-7 i \text { and } \Delta x=0.2$$

Use sigma notation to represent each sum. $$1+2+3+4+5+\cdots+21+22+23$$
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