Chapter 12: Q. 16 (page 838)

In Problems 13–16, find each sum.
$\sum_{k = 1}^{7} {(\frac{1}{3})}^{k}$

Short Answer

Expert verified

The sum is: $\frac{1}{2} (1 - 3^{- 7})$

Step by step solution

Step 1. Given information.

The summation notation of the sequence is:

$\sum_{k = 1}^{7} {(\frac{1}{3})}^{k}$

Step 2. Expand the summation by substituting values of k.

$\sum_{k = 1}^{7} {(\frac{1}{3})}^{k} = {(\frac{1}{3})}^{1} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{3} + {(\frac{1}{3})}^{4} + . . . . . . . {(\frac{1}{3})}^{7}$

Step 3. Check type of sequence.

Here we can see that ratio between successive term is constant.

So given series is a geometric series.

First term is $a = \frac{1}{3}$

common ratio is $r = {(\frac{1}{3})}^{2} \div (\frac{1}{3}) = \frac{1}{3}$

Number of terms is $n = 7$

Step 4. Find sum of geometric series.

The sum of all 7 terms:

\frac{1}{3} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{3} + . . . . . + {(\frac{1}{3})}^{7} = \frac{\frac{1}{3} \{1 - {(\frac{1}{3})}^{7}\}}{1 - \frac{1}{3}} = \frac{\frac{1}{3} (1 - 3^{- 7})}{\frac{2}{3}} = \frac{1}{2} (1 - 3^{- 7)}

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.

In Problems 13–16, find each sum.
$\sum_{k = 1}^{7} {(\frac{1}{3})}^{k}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Expand the summation by substituting values of k.

Step 3. Check type of sequence.

Step 4. Find sum of geometric series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Probability and Statistics

Applied Mathematics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 13–16, find each sum. ∑k=1713k

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Expand the summation by substituting values of k.

Step 3. Check type of sequence.

Step 4. Find sum of geometric series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Probability and Statistics

Applied Mathematics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 13–16, find each sum.
$\sum_{k = 1}^{7} {(\frac{1}{3})}^{k}$