Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 19: Problem 5

The sum to infinity of a geometric sequence is four times the first term. Find the common ratio.

Short Answer

Expert verified
Answer: The common ratio of this geometric sequence is 3/4.

Step by step solution

01

Setting up the equation

The exercise says that the sum to infinity (S) is four times the first term (a), so we can write: S = 4a. We know that the sum to infinity formula for a geometric sequence is S = a / (1 - r). Now, we can replace S in the formula with 4a: 4a = a / (1 - r)
02

Solve for the common ratio r

Now, we have the equation 4a = a / (1 - r). The next step is to isolate r: 4a(1 - r) = a 4 - 4r = 1 4r = 3 r = 3/4 So, the common ratio r is equal to 3/4.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the seventh term of a geometric sequence with first term 2 and common ratio 3 .

Use the power series expansion of \(\mathrm{e}^{x}\) to show that $$ \mathrm{e}^{2 x}=1+2 x+2 x^{2}+\frac{4 x^{3}}{3}+\cdots $$

A sequence is given by \(5, \frac{5}{8}, \frac{5}{27}, \frac{5}{64}, \ldots\) Write down an expression to denote the full sequence.

Write out explicitly the series $$ \sum_{k=1}^{4} \frac{1}{(2 k+1)(2 k+3)} $$

Find the sum of the first 40 positive integers.
See all solutions

Recommended explanations on Physics Textbooks

Solid State Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Waves Physics

Read Explanation

Medical Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Geometrical and Physical Optics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free