Problem 20 - Algebra 2 and Trigonometry

Question

In \(15-22 :\) a. Write each sum as a series. b. Find the sum of each series. $$ 1+\sum_{n=1}^{5}\left(\frac{2}{3}\right)^{n} $$

Step-by-Step Solution

Answer

The sum of the series is approximately 0.131.

Step 1: Understanding the Series

The given expression is \(1+\sum_{n=1}^{5}\left(\frac{2}{3}\right)^{n}\). This means we have a sum of 1 and a geometric series starting from \(n=1\) to \(n=5\) with the term \(\left(\frac{2}{3}\right)^{n}\).

Step 2: Writing the Series

To write out the series, list each term in the sum: \(1 + \left(\frac{2}{3}\right)^{1} + \left(\frac{2}{3}\right)^{2} + \left(\frac{2}{3}\right)^{3} + \left(\frac{2}{3}\right)^{4} + \left(\frac{2}{3}\right)^{5}\).

Step 3: Calculating Each Term

Calculate each term of the series:- \(\left(\frac{2}{3}\right)^{1} = \frac{2}{3}\)- \(\left(\frac{2}{3}\right)^{2} = \frac{4}{9}\)- \(\left(\frac{2}{3}\right)^{3} = \frac{8}{27}\)- \(\left(\frac{2}{3}\right)^{4} = \frac{16}{81}\)- \(\left(\frac{2}{3}\right)^{5} = \frac{32}{243}\)

Step 4: Summing the Series

Add up all the terms of the series:\[1 + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \frac{16}{81} + \frac{32}{243} = \text{Total Sum}\]

Step 5: Find the Sum of Geometric Series

Use the formula for the sum of a finite geometric series:\[ S_n = a \frac{r^n-1}{r-1} \]Where \(a = \frac{2}{3}\) (first term), \(r = \frac{2}{3}\) (common ratio), and \(n = 5\) (number of terms):\[ S_5 = \frac{2}{3} \frac{(\frac{2}{3})^5 - 1}{\frac{2}{3} - 1} = \frac{2}{3} \left(\frac{\frac{32}{243} - 1}{\frac{-1}{3}}\right) = \frac{2}{3} \left(\frac{-211}{243} \right) \]Now add \(1\) to the sum of the series approximately to find total sum.

Step 6: Calculate the Total Sum Approximatively

Approximating the sum:1. Calculate the sum \( \frac{2}{3} \cdot \left(-\frac{211}{243} \right) \approx -0.869\)2. Add 1 to the sum of the geometric series: \[1 - 0.869 \approx 0.131\]

Step 7: Final Answer

The sum of the series is approximately 0.131.

Key Concepts

Sum of Series

When dealing with a series, our goal is often to find a single number that represents the addition of its terms. This single number is called the "sum of the series." In mathematical terms, a series is the sum of the terms of a sequence. In our exercise, we specifically deal with a geometric series that combines multiple exponential terms.

• The sum could be finite or infinite, but in this case, we're dealing with a finite series bound by a specific number of terms.
• The series uses terms that follow a specific rule or pattern - here, each term is a power of the common ratio.

Breaking down the given exercise, the sum resulting from adding the terms from 1 to a geometric sequence is computed using a series that consists of five terms raised to specific powers of the common ratio \(\left(\frac{2}{3}\right)\). Understanding how to correctly evaluate these terms and their sum harnesses our understanding of how series work in mathematics.

Finite Geometric Series

A finite geometric series is a series with a limited number of terms, as opposed to an infinite geometric series, which continues indefinitely. The distinctive feature of a geometric series, whether finite or infinite, is that each term is obtained by multiplying the previous term by a constant factor called the common ratio.

• To find the sum of a finite geometric series, one can use the formula: \( S_n = a \frac{r^n - 1}{r - 1} \).
• In this formula, \(a\) represents the first term of the series, \(r\) is the common ratio, and \(n\) is the number of terms.

In our given example, we have the first term \(a = \frac{2}{3}\), a common ratio \(r = \frac{2}{3}\), and 5 terms. Plugging into the formula allows us to find the precise sum of the sequence before any adjustments or approximations are made.

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The backbone of solving any geometric series problem is understanding the sequence involved.

• To break it down, if our first term is \(a\), then the second term becomes \(ar\), the third is \(ar^2\), and so on.
• In our series \( 1+ \sum_{n=1}^{5}\left(\frac{2}{3}\right)^n \), the geometric sequence part is the list \(\left(\frac{2}{3}\right)^n\) for \(n = 1\) to \(5\).

Recognizing this pattern enables you to write the terms of the geometric series completely and accurately, laying the foundation for calculating their sum.

Common Ratio

The common ratio in a geometric sequence or series is the factor by which we multiply a term to get the next term. It holds everything together in a geometric sequence.

• This multiplier is crucial because it dictates the exponential growth (or decay) of the series.
• In our problem, the common ratio is consistently \(\frac{2}{3}\), impacting how each subsequent term is calculated as \(\left(\frac{2}{3}\right)^{n}\).

Since each term is derived from this common ratio, understanding it aids in capturing the essence of what makes a sequence geometric. Calculations for both series content and their cumulative sum are directly influenced by its value, highlighting its importance in solving geometric series problems.