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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q. 11 (page 816)

In the problem, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\{t_{n}\} = \{\frac{1}{2} - \frac{1}{3} n\}$

Short Answer

Expert verified

The solution is $d = - \frac{1}{3}$ ; first four terms $= \frac{1}{6}, - \frac{1}{6}, - \frac{1}{2}, - \frac{5}{6}$ .

Step by step solution

01

Step 1. Given information

$\{t_{n}\} = \{\frac{1}{2} - \frac{1}{3} n\}$

02

Step 2. Calculation

The first term is $t_{1} = \frac{1}{2} - \frac{1}{3} 路 1 = \frac{1}{6}$ .

The $n$ th term and the $(n - 1)$ st term of the sequence are-

$t_{n} = \frac{1}{2} - \frac{1}{3} n$ and $t_{n - 1} = \frac{1}{2} - \frac{1}{3} (n - 1) = \frac{1}{2} - \frac{1}{3} n + \frac{1}{3} = \frac{5}{6} - \frac{1}{3} n$

The common difference is -

$d = t_{n} - t_{n - 1} = (\frac{1}{2} - \frac{1}{3} n) - (\frac{5}{6} - \frac{1}{3} n) = - \frac{1}{3}$

Since the difference of any two successive terms is the constant $- \frac{1}{3}$ , the sequence $\{t_{n}\}$ is arithmetic.

The first four terms are $\frac{1}{6}, - \frac{1}{6}, - \frac{1}{2}, - \frac{5}{6}$ .

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Most popular questions from this chapter

In Problems 37鈥�50, a sequence is defined recursively. Write down the first five terms.

$a_{1} = - 2; a_{n} = n + 3 a_{n - 1}$

A(n) _______ is a function whose domain is the set of positive integers.

In Problems 11鈥�16, evaluate each factorial expression.

$\frac{12!}{10!}$

In Problems 61鈥�70, express each sum using summation notation.

$1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + . . . . . . + {(- 1)}^{6} \frac{1}{3^{6}}$

In Problems 17鈥�28, write down the first five terms of each sequence.

$\{s_{n}\} = \{n\}$

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