Chapter 12: Q. 9 (page 838)

In Problems 7–12, 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.
$\{s_{n}\} = \{2^{3 n}\}$

Short Answer

Expert verified

Given sequence is a geometric sequence.
The common ratio of the sequence is 8.
Th sum of first $n$ terms is: $S_{n} = \frac{8}{7} (8^{n} - 1)$

Step by step solution

Step 1. Given information

Sequence: $\{s_{n}\} = \{2^{3 n}\}$

Step 2. To find first term and n-1th term.

$s_{1} = 2^{3.1} = 2^{3} = 8$

$s_{n - 1} = 2^{3 (n - 1)} = 2^{3 n - 3} = \frac{2^{3 n}}{2^{3}}$

Step 3. To find ratio between two successive terms.

Ratio between two successive terms is:

$r = \frac{s_{n}}{s_{n - 1}} = \frac{2^{3 n}}{\frac{2^{3 n}}{2^{3}}} = 2^{3} = 8$

Since ratio is constant.

Hence the given sequence is a geometric sequence whose common ratio is 8.

Step 4. To find sum of first n term of the sequence.

$S_{n} = \frac{8 (8^{n} - 1)}{8 - 1} S_{n} = \frac{8}{7} (8^{n} - 1)$

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In Problems 7–12, 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.
$\{s_{n}\} = \{2^{3 n}\}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. To find first term and n-1th term.

Step 3. To find ratio between two successive terms.

Step 4. To find sum of first n term of the sequence.

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In Problems 7–12, 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.sn=23n

Short Answer

Step by step solution

Step 1. Given information

Step 2. To find first term and n-1th term.

Step 3. To find ratio between two successive terms.

Step 4. To find sum of first n term of the sequence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 7–12, 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.
$\{s_{n}\} = \{2^{3 n}\}$