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Chapter 10: Problem 88

This set of exercises will draw on the ideas presented in this section and your general math background. Suppose \(a, b,\) and \(c\) are three consecutive terms in an arithmetic sequence. Show that \(b=\frac{a+c}{2}\)

Short Answer

Expert verified
The expression \(b = \frac{a + c}{2}\) is proven by the property of an arithmetic progression that the difference between any two successive terms is a constant. Applying this property, we get the required expression after simplifying the equation.

Step by step solution

01

Understand the Property of Arithmetic Progression

The first step is to understand a key property of an arithmetic progression: the difference between any two successive terms is constant. Meaning for our sequence \(a, b, c\), the difference \(b-a\) should equal the difference \(c-b\).
02

Formulate the equation

Using this property, we can form an equation. Given \(b-a = c-b\), we can re-arrange the terms to isolate \(b\) on one side.
03

Solve the equation

To isolate \(b\), we add \(b\) and subtract \(a\) from both sides which gives us \(2b = a + c\), and thus, dividing the whole equation by 2, we get \(b = \frac{a + c}{2}\). This is the required expression.

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Key Concepts

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

Consecutive Terms in an Arithmetic Sequence
An arithmetic sequence is a series of numbers where each term after the first is derived by adding a constant, known as the 'common difference', to the previous term. The term "consecutive" refers to terms that are next to each other in this sequence.
Consider expressing these consecutive terms: if the first term is denoted by \(a\), the next one which would increase by the common difference \(d\) is \(a + d\), followed by \(a + 2d\), and so on.
In our specific problem, \(a, b,\) and \(c\) represent three consecutive terms. This means:
  • \(b = a + d\)
  • \(c = a + 2d\)
Understanding how consecutive terms work is essential to solve complex problems around arithmetic sequences.
Understanding the Difference of Terms
In an arithmetic sequence, the difference between any two successive terms is consistent and is known as the 'common difference'. This property underpins the identity of arithmetic sequences.
When we have terms such as \(a\), \(b\), and \(c\) in an arithmetic sequence, this leads us to the equation: \(b - a = c - b\). This equation showcases how the difference between terms remains equal.
Given the sequences like \(a, a+d,\) and \(a+2d\), the difference \(b-a\) simplifies to \(d\), and the same goes for \(c-b\). This balance in terms helps pave the way to find expressions that define arithmetic sequences easier to work with in algebraic form.
Isolating Variables to Express Terms
In algebra, isolating variables is crucial in solving equations. It stands for rearranging an equation to express one variable explicitly in terms of others. This allows for an easier understanding and solving of algebraic problems.
In the context of our task, starting from the equation \(b-a = c-b\), isolating \(b\) involves a series of steps:
  • Add \(b\) to both sides: \(b-a + b = c\)
  • Reorganize: \(2b = a + c\)
  • Finally, divide by 2 to solve for \(b\): \(b = \frac{a+c}{2}\)

Through the manipulation of this expression, we achieve a succinct formula that describes \(b\) as the average of \(a\) and \(c\). This improves our understanding, as it clarifies the relationships and dependencies among variables used in arithmetic progressions.

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Most popular questions from this chapter

In this set of exercises, you will use sequences and their sums to study real- world problems. The following pocm (As IWas Going to St. Ives, circa 1730 ) refers to the name of a quaint old village in Cornwall, England. As I was going to St. Ives I met a man with seven wives. Every wife had seven sacks, Every sack had seven cats, Every cat had seven kits. Kits, cats, sacks, and wives, How many were going to St. Ives? (a) Use the sum of a sequence of numbers to express the number of people and objects (combined) that the author of this poem encountered while going to St. Ives. Do not evaluate the sum. Is this the sum of terms of an arithmetic sequence or a geometric sequence? Explain. (b) Use an appropriate formula to find the sum from part (a).

State whether the sequence is arithmetic or geometric. $$-7,-11,-15,-19, \dots$$

Use counting principles from Section 10.4 to calculate the number of outcomes. A group of friends, five girls and five boys, wants to go to the movies on Friday night. The friends select, at random, two of their group to go to the ticket office to purchase the tickets. What is the probability that the two selected are both boys?

In this set of exercises, you will use sequences to study real-world problems. An employee starting with an annual salary of \(\$ 40,000\) will receive a salary increase of \(4 \%\) at the end of each year. What type of sequence would you use to find her salary after 6 years on the job? What is her salary after 6 years?

In this set of exercises, you will use sequences and their sums to study real- world problems. A carpet warehouse needs to calculate the diameter of a rolled carpet given its length, width, and thickness. If the diameter of the carpet roll can be predicted ahead of time, the warehouse will know how much to order so as not to exceed warehouse capacity. Assume that the carpet is rolled lengthwise. The crosssection of the carpet roll is then a spiral. To simplify the problem, approximate the spiral cross-section by a set of \(n\) concentric circles whose radii differ by the thickness \(t\) Calculate the number of circles \(n\) using the fact that the sum of the circumferences of the \(n\) circles must equal the given length. How can you find the diameter once you know \(n ?\)
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