Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

4

Expert-verified

Abstract Algebra: An Introduction

Found in: Page 528

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

Let r be a real number, $r \neq 1$ . Prove that for every integer $n \geq 1$ , $1 + r + r^{2} + r^{3} + . . . . + r^{n - 1} = \frac{r^{n} - 1}{r - 1}$
.

It is proved that for every integer $n \geq 1$ :

$1 + r + r^{2} + r^{3} + . . . . + r^{n - 1} = \frac{r^{n} - 1}{r - 1}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Consider the given parameters

Assume that $r \neq 1$ is a real number. The objective is to show that for each integer $n \geq 1$ .

$1 + r + r^{2} + r^{3} + . . . . + r^{n - 1} = \frac{r^{n} - 1}{r - 1}$

Obtain the equation for Sr

Let $S = 1 + r + r^{2} + r^{3} + . . . + r^{n - 1}$ …. (1)

Then multiply both the sides of the sum S by the real number r and get the equation as follows:

role="math" localid="1648726203258" $S r = r (1 + r + r^{2} + r^{3} + . . . . r^{n - 1}) S r = r + r^{2} + r^{3} + . . . . . + r^{n - 1} + r^{n} . . . . . (2)$

Solve the equations (1) and (2)

Subtract equation (1) from equation (2) and solve in the following manner:

$\begin{array}{rcl} r S - S & = & r + r^{2} + r^{3} + . . . . + r^{n - 1} + r^{n} - (1 + r + r^{2} + r^{3} + . . . . r^{n - 1}) \\ (r - 1) S & = & r + r^{2} + r^{3} + . . . . + r^{n - 1} + r^{n} - 1 - r - r^{2} - r^{3} - . . . . . - r^{n - 1} \\ (r - 1) S & = & r^{n} - 1 \\ S & = & \frac{r^{n} - 1}{r - 1} \end{array}$

$S = 1 + r + r^{2} + r^{3} + . . . . + r^{n - 1}$ ; therefore, $1 + r + r^{2} + r^{3} + . . . . . . . + r^{n - 1} = \frac{r^{n} - 1}{r - 1}$ .

Most popular questions for Math Textbooks

Let B be a finite set and $f : B \to B$ is a function. Prove that f is injective if and only if f is surjective.

Let A be a set and ${A_{i} | i \in I}$ a partition of A. Define a relation on A by: $a ~ b$ if and only if a and b are in the same subset of the partition (that is, there exists $k \in l$ such that role="math" localid="1659180795678" $a \in A_{k}$ and $b \in A_{k}$ .

(a) Prove that - is an equivalence relation on A.

(b) Prove that the equivalence classes of $~$ are precisely the subsets $A_{i}$ of the partition.

Describe each set by listing:

(a) The integers strictly between -3 and 9.

(b) The negative integers greater than -10.

(c) The positive integers whose square roots are less than or equal to 4.

NOTE: Z is the set of integers, Q is the set of rational numbers, and Z the set of real numbers.

Let P be a plane. If p, q are points in P, then $p ~ q$ means p and q are the same distance from the origin. Prove that $~$ is an equivalence relation on P.

Question: Let r and n be integers with 0 < r < n. Prove that $[\begin{matrix} n \\ r \end{matrix}] = [\begin{matrix} n \\ n - r \end{matrix}]$ .

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free