Chapter 12: Q 39. (page 825)

In Problems 39– 44, find each sum.
$\frac{1}{4} + \frac{2}{4} + \frac{2^{2}}{4} + \frac{2^{3}}{4} + . . . + \frac{2^{14}}{4}$

Short Answer

Expert verified

The sum of the given sequence is $\frac{1}{4} \cdot - (1 - 2^{n})$

Step by step solution

Step 1. Write the given information.

The given sequence is $\frac{1}{4} + \frac{2}{4} + \frac{2^{2}}{4} + \frac{2^{3}}{4} + . . . + \frac{2^{14}}{4}$

Step 2. Find the common ratio.

$a_{1} = \frac{1}{4}, a_{2} = \frac{3}{4}, a_{3} = \frac{2^{2}}{4} \frac{a_{2}}{a_{1}} = \frac{\frac{2}{4}}{\frac{1}{4}} = 2 \frac{a_{3}}{a_{2}} = \frac{\frac{2^{2}}{4}}{\frac{2}{4}} = 2 r = 2$

Step 3. Find the sum of nth term.

The formula for the sum of nth terms $s_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}$

$s_{n} = \frac{1}{4} \cdot \frac{1 - 2^{n}}{1 - 2} = \frac{1}{4} \cdot - (1 - 2^{n})$

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In Problems 39– 44, find each sum.
$\frac{1}{4} + \frac{2}{4} + \frac{2^{2}}{4} + \frac{2^{3}}{4} + . . . + \frac{2^{14}}{4}$

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Find the common ratio.

Step 3. Find the sum of nth term.

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In Problems 39– 44, find each sum.14+24+224+234+...+2144

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Find the common ratio.

Step 3. Find the sum of nth term.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

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Pure Maths

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Study anywhere. Anytime. Across all devices.

In Problems 39– 44, find each sum.
$\frac{1}{4} + \frac{2}{4} + \frac{2^{2}}{4} + \frac{2^{3}}{4} + . . . + \frac{2^{14}}{4}$