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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. 8 (page 838)

In Problems 7鈥�12, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms.
$\{c_{n}\} = \{2 n^{3}\}$

Short Answer

Expert verified

Given sequence is neither arithmetic nor geometric sequence.

Step by step solution

01

Step 1. Given information.

Sequence is:

$\{c_{n}\} = \{2 n^{3}\}$

02

Step 2. Find the first term and n-1th term of the sequence.

$\begin{array}{rcl} c_{1} & = & 2 {(1)}^{3} \\ = & 2 \end{array}$

$\begin{array}{rcl} c_{n - 1} & = & 2 {(n - 1)}^{3} \\ = & 2 (n^{3} - 1 - 3 n^{2} + 3 n) \end{array}$

03

Step 3. Check the difference between two successive terms.

Difference between two successive terms:

d = c_{n} - c_{n - 1} = 2 n^{3} - 2 (n^{3} - 1 - 3 n^{2} + 3 n) = 2 n^{3} - 2 n^{3} + 2 + 6 n^{2} - 6 n = 2 + 6 n^{2} - 6 n

Since difference is not constant, hence the given sequence is not arithmetic sequence.

04

Step 4. Check the ratio between two successive terms.

Ratio between two successive terms:

$r = \frac{c_{n}}{c_{n - 1}} = \frac{2 n^{3}}{2 (n^{3} - 1 - 3 n^{2} + 3 n)} = \frac{n^{3}}{(n^{3} - 1 - 3 n^{2} + 3 n)}$

Since ratio is also not constant.

Hence given sequence is not a geometric sequence.

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Most popular questions from this chapter

In Problems 29鈥�36, the given pattern continues. Write down the nth term of a sequence $\{a_{n}\}$ suggested by the pattern.

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