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Chapter 11: Problem 93

What is the difference between a geometric sequence and an infinite geometric series?

Short Answer

Expert verified
A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero number while an infinite geometric series is the sum of an infinite geometric sequence. An infinite geometric series will converge to a finite number if the absolute value of the common ratio is less than one.

Step by step solution

01

Define the Geometric Sequences

A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, \(2, 6, 18, 54, ...\). The general form of a geometric sequence is \(a, ar, ar^2, ar^3, ...\) where 'a' is the first term (non-zero), 'r' is the common ratio (non-zero).
02

Define the Infinite Geometric Series

An infinite geometric series is the sum of the terms of an infinite geometric sequence. It is expressed as \(a + ar + ar^2 + ar^3 + ...\). An infinite geometric series has a sum if the common ratio 'r' is between -1 and 1. The sum (S) of an infinite geometric series can be calculated by the formula \(S = a / (1 - r)\).
03

Differentiate between Geometric Sequences and Infinite Geometric Series

Geometric sequence refers to the list of numbers in order where each term is obtained by multiplying the previous one by a constant ratio, whereas an infinite geometric series is the sum of the terms of this sequence, with the condition that the absolute value of the common ratio should be less than 1. In other words, a geometric sequence focuses on the individual terms while an infinite geometric series focuses on the sum of these terms.

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Most popular questions from this chapter

Use the formula for the value of an annuity to solve Exercises 77鈥�84. Round answers to the nearest dollar. To save money for a sabbatical to earn a master's degree, you deposit \(\$ 2000\) at the end of each year in an annuity that pays \(7.5 \%\) compounded annually. a. How much will you have saved at the end of five years? b. Find the interest.

A fair coin is tossed two times in succession. The sample space of equally likely outcomes is \(\\{H H, H T, T H, T T\\} .\) Find the probability of getting $$\text{two heads.}$$

Use this information to solve Exercises \(47-48 .\) The mathematics department of a college has 8 male professors, 11 female professors, 14 male teaching assistants, and 7 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is a professor or a female.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Beginning at 6: 45 A.M., a bus stops on my block every 23 minutes, so I used the formula for the \(n\) th term of an arithmetic sequence to describe the stopping time for the \(n\) th bus of the day.

Write an equation in point-slope form and slope-intercept form for the line passing through \((-2,-6)\) and perpendicular to the line whose equation is \(x-3 y+9=0 .\) (Section 2.4 Example \(2)\)
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