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Chapter 15: Problem 64

Use the formula for \(S_{n}\) to find the sum of the terms of each geometric sequence. $$\sum_{i=1}^{5} 2\left(\frac{1}{3}\right)^{i}$$

Short Answer

Expert verified
The sum of the first 5 terms of the geometric series \(\sum_{i=1}^{5} 2\left(\frac{1}{3}\right)^{i}\) is \(S_5 = \frac{726}{243}\).

Step by step solution

01

Identify the geometric series and the common ratio

We are given the geometric series: $$\sum_{i=1}^{5} 2\left(\frac{1}{3}\right)^{i}$$ From the expression, we can see that the first term of the series, a, is 2 and the common ratio, r, is \(\frac{1}{3}\).
02

Recall the formula for the sum of a geometric series

The formula for the sum of the first n terms of a geometric series with the first term a and common ratio r is as follows: $$S_n = \frac{a\left(1-r^n\right)}{1-r}$$ In this exercise, we are asked to find the sum of the first 5 terms, so n = 5.
03

Substitute the values into the formula

Now, we will substitute the values of a, r, and n into the formula for the sum of the geometric series: $$S_5 = \frac{2\left(1-\left(\frac{1}{3}\right)^{5}\right)}{1-\frac{1}{3}}$$
04

Simplify the expression

Simplify the expression to find the sum of the series: $$S_5 = \frac{2\left(1-\left(\frac{1}{3}\right)^{5}\right)}{\frac{2}{3}} \cdot \frac{3}{2}$$ $$S_5 = 3\left(1-\left(\frac{1}{3}\right)^{5}\right)$$ $$S_5 = 3\left(1-\frac{1}{243}\right)$$
05

Calculate the sum

Finally, calculate the sum of the first 5 terms of the geometric series: $$S_5 = 3\left(\frac{242}{243}\right)$$ $$S_5 = \frac{726}{243}$$ So, the sum of the first 5 terms of the given geometric series is \(\frac{726}{243}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Geometric Series
In the world of sequences and series, a geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, called the common ratio. To find the sum of a geometric series, we use a special formula. This formula is very powerful as it allows us to sum terms without manually adding each one. For the sum of the first \( n \) terms of a geometric series with the first term \( a \) and common ratio \( r \), the formula is:
  • \( S_n = \frac{a(1 - r^n)}{1 - r} \)
This formula is crucial when dealing with problems involving many sequential calculations. Using it, you can quickly find out how many scoops make up your ice cream stack of numbers!
Common Ratio
The common ratio is the backbone of a geometric sequence. It's the number you consistently multiply by to get from one term to the next. Imagine it like the secret ingredient in a recipe that transforms each ingredient to the next stage. In our series \( \sum_{i=1}^{5} 2\left(\frac{1}{3}\right)^{i} \), the common ratio is \( \frac{1}{3} \). This means, starting from the first term, each subsequent term is a third of the previous one. Understanding the common ratio allows you to predict how any term will shift in a series even before performing any calculations.
Geometric Sequence
Geometric sequences are like building blocks. They start with a single term and build upon it using a repetitive process. Each term in a geometric sequence is the result of multiplying the prior term by a specific number, known as the common ratio. For example, in our series, the first term is 2, then the next term is \( 2 \times \frac{1}{3} \), and so on.
  • The sequence would appear as: 2, \( \frac{2}{3} \), \( \frac{2}{9} \), \( \frac{2}{27} \), \( \frac{2}{81} \).
Visualizing these sequences is much like arranging a lineup of dominoes; each one is set by the fall of the previous.
Series Formula
To really grasp the beauty of geometric series, knowing how to use the series formula is essential. Once you have identified your first term \( a \) and common ratio \( r \), you plug these into the series formula. For our given series, this involves calculating:
  • First, solving for \( r^n \): \( \left(\frac{1}{3}\right)^5 \).
  • Then, using the formula: \( S_5 = \frac{2(1 - \left(\frac{1}{3}\right)^5)}{1-\frac{1}{3}} \).
  • Simplify your result: \( S_5 = \frac{726}{243} \).
This formula not only simplifies complex calculations but also helps you explore patterns in different series. With practice, you'll see how powerful and versatile this equation is in unlocking the mysteries of numbers.

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Most popular questions from this chapter

Use the binomial theorem to expand each expression. $$(t+9)^{3}$$

Use the binomial theorem to expand each expression. $$(2 k+1)^{4}$$

Find the sum of the terms of the infinite geometric sequence, if possible. $$a_{1}=20, r=-\frac{3}{4}$$

A ball is dropped from a height of 27 ft. Each time the ball bounces it rebounds to \(\frac{2}{3}\) of its previous height. a) Find the height the ball reaches after the fifth bounce. b) Find the total vertical distance the ball has traveled when it comes to rest. (GRAPH CANT COPY)

Find the sum of the terms of the infinite geometric sequence, if possible. $$a_{1}=18, r=\frac{1}{3}$$
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