Chapter 12: Q. 12.21 (page 1171)

Write the sum using summation notation: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} .$

Short Answer

Expert verified

The sum of $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32}$ can be written in the summation notation as $\sum_{n = 1}^{5} \frac{1}{2^{n}} .$

Step by step solution

Step 1. Given information

$\begin{array}{l} Thegivensumis \\ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} \\ Wehavetowritethesumusingsummationnotation. \end{array}$

Step 2. Find the pattern

Now,

We look for a pattern in the terms.

$\begin{matrix} n : & 1 & 2 & 3 & 4 & 5 & ... & n \\ Terms & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} & \frac{1}{16} & \frac{1}{32} & ... \\ Pattern & \frac{1}{2} & \frac{1}{2^{2}} & \frac{1}{2^{3}} & \frac{1}{2^{4}} & \frac{1}{2^{5}} & ... & \frac{1}{2^{n}} \end{matrix}$

Step 3. Conclusion

From the above table, we can observe that all the numerators are 1 and the denominators are 2 raised to the term.

Therefore,

We can write the sum for the 5 terms in summation notation as

$\sum_{n = 1}^{5} \frac{1}{2^{n}} .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Write the sum using summation notation: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the pattern

Step 3. Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Probability and Statistics

Logic and Functions

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

Write the sum using summation notation:12+14+18+116+132.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the pattern

Step 3. Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Probability and Statistics

Logic and Functions

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

Write the sum using summation notation: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} .$