Chapter 7: Q. 29 (page 592)

In Exercises, find a plausible formula for the general term of the given sequence.
$\{\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \dots\}$

Short Answer

Expert verified

The plausible formula for the general term of the sequence is $a_{k} = \frac{k + 1}{2 k + 1}$ where $k \geq 1 .$

Step by step solution

Step 1. Given information.

The given sequence is $\{\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \dots\} .$

Step 2. plausible formula for the general term.

write the sequence in a pattern.

$\{\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \dots\} = \{\frac{1 + 1}{2 (1) + 1}, \frac{2 + 1}{2 (2) + 1}, \frac{3 + 1}{2 (3) + 1}, \frac{4 + 1}{2 (4) + 1}, \dots\} \{\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \dots\} = {\{\frac{k + 1}{2 k + 1}\}}_{k = 1}^{\infty}$

for every term, the numerator is $(k + 1)$ and the denominator is $(2 k + 1)$ andlocalid="1649189107593" $k \geq 1 .$

So plausible formula for the general term of the sequence islocalid="1649189440639" $a_{k} = \frac{k + 1}{2 k + 1} .$

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In Exercises, find a plausible formula for the general term of the given sequence.
$\{\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \dots\}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. plausible formula for the general term.

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In Exercises, find a plausible formula for the general term of the given sequence.23,35,47,59,…

Short Answer

Step by step solution

Step 1. Given information.

Step 2. plausible formula for the general term.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Mechanics Maths

Applied Mathematics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Exercises, find a plausible formula for the general term of the given sequence.
$\{\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \dots\}$