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Chapter 14: Problem 16

Find the first term and the common difference. $$ \$ 214, \$ 211, \$ 208, \$ 205, \dots $$

Short Answer

Expert verified
First term is \$214 and the common difference is \(-3\).

Step by step solution

01

Identify the Sequence

The given sequence is: \[\$214, \$211, \$208, \$205, \dots\]This sequence is an arithmetic sequence because each term is decreased by the same amount.
02

Find the First Term

The first term of the sequence is just the first number provided in the series.\[a_1 = \$214\]
03

Calculate the Common Difference

To find the common difference (\[d\]), subtract the second term from the first term.\[d = 211 - 214 = -3\]So, the common difference is \(-3\).

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Key Concepts

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

first term
In an arithmetic sequence, the first term is crucial. It sets the tone for the entire sequence. First term is denoted by \(a_1\). In the given sequence, \[214, 211, 208, 205, \dots \], 214 is the starting point. So, \(a_1 = \$214\). This means every term in the sequence builds upon this initial value.
common difference
The common difference is the value that is added (or subtracted) each time to get from one term to the next in an arithmetic sequence. In the given sequence, \[214, 211, 208, 205, \dots \], each term decreases by 3 dollars. To find the common difference \(d\), subtract the second term from the first term: \[d = 211 - 214 = -3\]. So, here the common difference, \(d\), is \(-3\). This negative value shows that the sequence is decreasing.
series
When we talk about an arithmetic series, we're referring to the sum of the terms of an arithmetic sequence. For example, in our sequence \[214, 211, 208, 205, \dots \], if we want to find the sum of the first few terms, we would add them together: \(214 + 211 + 208 + 205\). The formula to find the sum of the first \(n\) terms \(S_n\) of an arithmetic series is: \[S_n = \frac{n}{2}\cdot(2a_1 + (n-1)\cdot d)\].
Using this formula ensures quick calculation, without needing to sum each term individually.

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Most popular questions from this chapter

Straight-Line Depreciation. A company buys a color laser printer for \(\$ 5200\) on January 1 of a given year. The machine is expected to last for 8 years, at the end of which time its trade-in, or salvage, value will be \(\$ 1100 .\) If the company figures the decline in value to be the same each year, then the trade-in values, after \(t\) years, \(0 \leq t \leq 8,\) form an arithmetic sequence given by $$ a_{t}=C-t\left(\frac{C-S}{N}\right) $$ where \(C\) is the original cost of the item, \(N\) the years of expected life, and \(S\) the salvage value. a) Find the formula for \(a_{t}\) for the straight-line depreciation of the printer. b) Find the salvage value after 0 year, 1 year, 2 years, 3 years, 4 years, 7 years, and 8 years. c) Find a formula that expresses \(a_{t}\) recursively.

Find a formula for the sum of the first \(n\) consecutive odd numbers starting with 1: $$ 1+3+5+\cdots+(2 n-1) $$

Show that there are exactly \(\left(\begin{array}{l}{5} \\\ {3}\end{array}\right)\) ways of choosing a subset of size 3 from \(\\{a, b, c, d, e\\}\).

Simplify. $$ \left(a_{1}+5 d\right)+\left(a_{n}-5 d\right) $$

Rewrite each sum using sigma notation. Answers may vary. $$ \frac{1}{1 \cdot 2^{2}}+\frac{1}{2 \cdot 3^{2}}+\frac{1}{3 \cdot 4^{2}}+\frac{1}{4 \cdot 5^{2}}+\cdots $$
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