Chapter 12: Q 11. (page 824)

Show that each sequence is geometric. Then find the common ratio and write out the first four terms $\{c_{n}\} = \{\frac{2^{n - 1}}{4}\}$ .

Short Answer

Expert verified

The common ratio is $2$ and the first four terms are $\frac{1}{4}, \frac{1}{2}, 1, 2$ .

Step by step solution

Step 1. Write the given information.

We have given sequence.

$\{c_{n}\} = \{\frac{2^{n - 1}}{4}\}$

The nth and (n-1)th term are

c_{n} = \frac{2^{n - 1}}{4} c_{n - 1} = \frac{2^{(n - 1) - 1}}{4} c_{n - 1} = \frac{2^{n - 2}}{4}

Step 2. To find the common ratio.

We have,

$r = \frac{c_{n}}{c_{n - 1}} r = \frac{\frac{2^{n - 1}}{4}}{\frac{2^{n - 2}}{4}} r = 2^{n - 1 - n + 2} r = 2$

Since the ratio of the successive term is the non zero constant $2$ , therefore $\{c_{n}\}$ is a geometric sequence with common ratio is $2$ .

Step 3. To find the terms.

We have,

$c_{1} = \frac{2^{1 - 1}}{4} c_{1} = \frac{1}{4} c_{2} = \frac{2^{2 - 1}}{4} c_{2} = \frac{2}{4} c_{2} = \frac{1}{2} c_{3} = \frac{2^{3 - 1}}{4} c_{3} = \frac{4}{4} c_{3} = 1 c_{4} = \frac{2^{4 - 1}}{4} c_{4} = \frac{8}{4} c_{4} = 2$

Thus, the first four terms are $\frac{1}{4}, \frac{1}{2}, 1, 2$

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Show that each sequence is geometric. Then find the common ratio and write out the first four terms $\{c_{n}\} = \{\frac{2^{n - 1}}{4}\}$ .

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. To find the common ratio.

Step 3. To find the terms.

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Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. To find the common ratio.

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One App. One Place for Learning.

Most popular questions from this chapter

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Show that each sequence is geometric. Then find the common ratio and write out the first four terms $\{c_{n}\} = \{\frac{2^{n - 1}}{4}\}$ .