Question

Find each sum. Suppose that $1 / a, 1 / b,$ and $1 / c$ are three consecutive terms in an arithmetic sequence. Show that (a) $\frac{a}{c}=\frac{a-b}{b-c}$ (b) $b=\frac{2 a c}{a+c}$

Name: Problem 38, Chapter 14: Precalculus
Uploaded: 2020-07-13T09:08:45-08:00
Duration: 4 min 1 s
Channel: Zachary Mitchell
Description: Find each sum. Suppose that 1 / a, 1 / b, and 1 / c are three consecutive terms in an arithmetic sequence. Show that (a) (a)/(c)=(a-b)/(b-c) (b) b=(2 a c)/(a+c)

   Find each sum.
 Suppose that $1 / a, 1 / b,$ and $1 / c$ are three consecutive terms in an arithmetic sequence. Show that
(a) $\frac{a}{c}=\frac{a-b}{b-c}$
(b) $b=\frac{2 a c}{a+c}$

Precalculus

David Cohen,… 7th Edition

Chapter 14, Problem 38

Instant Answer

Solved by Expert Zachary Mitchell

07/13/2020

Step 1

This means that the difference between $1/c$ and $1/b$ is equal to the difference between $1/b$ and $1/a$. We can write this as: \[1/c - 1/b = 1/b - 1/a\] Show more…

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Find each sum. Suppose that $1 / a, 1 / b,$ and $1 / c$ are three consecutive terms in an arithmetic sequence. Show that (a) $\frac{a}{c}=\frac{a-b}{b-c}$ (b) $b=\frac{2 a c}{a+c}$

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Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is constant. This property allows for the creation of linear equations that relate the terms, which is essential in deriving the relationships between reciprocals in the given problem.

Harmonic Sequence

A harmonic sequence is formed by taking the reciprocals of an arithmetic sequence. In other words, if the reciprocals of three numbers form an arithmetic sequence, then those numbers are in harmonic progression. Understanding this connection is key to analyzing the relationships provided in the problem.

Algebraic Manipulation

Algebraic manipulation encompasses techniques such as cross-multiplication and rearrangement of equations to simplify expressions and isolate variables. These techniques are crucial for proving equalities and obtaining the desired relations between the terms in the sequence.

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