Problem 49 - Algebra and Trigonometry

Question

Use a graphing utility to find the sum of each geometric sequence. $$ \sum_{n=1}^{15}\left(\frac{2}{3}\right)^{n} $$

Step-by-Step Solution

Answer

The sum of the geometric sequence is approximately 1.9947.

Step 1: Identify the first term and common ratio

In the geometric sequence \(\sum_{n=1}^{15}\left(\frac{2}{3}\right)^{n}\), the first term, \(a_1\), is \(\left(\frac{2}{3}\right)^1 = \frac{2}{3}\), and the common ratio, \(r\), is \(\frac{2}{3}\).

Step 2: Use the formula for the sum of the first n terms of a geometric sequence

The formula for the sum of the first \(n\) terms of a geometric sequence is \[S_n = a_1 \frac{1-r^n}{1-r}\] where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 3: Substitute the values into the sum formula

Substitute \(a_1 = \frac{2}{3}\), \(r = \frac{2}{3}\), and \(n = 15\) into the formula: \[S_{15} = \frac{2}{3} \frac{1-\left(\frac{2}{3}\right)^{15}}{1-\frac{2}{3}}\]

Step 4: Simplify the expression

Simplify the denominator \((1 - \frac{2}{3})\) to get \((\frac{1}{3})\). The expression now looks like this: \[S_{15} = \frac{2}{3} \times \frac{1-\left(\frac{2}{3}\right)^{15}}{\frac{1}{3}}\]

Step 5: Multiply inside the fraction

Multiply \(\frac{2}{3}\) by the reciprocal of \(\frac{1}{3}\), which is 3: \[S_{15} = \frac{2}{3} \times 3 \times \left(1-\left(\frac{2}{3}\right)^{15}\right) = 2 \times \left(1-\left(\frac{2}{3}\right)^{15}\right)\]

Step 6: Compute the power and subtraction

Calculate \(\left(\frac{2}{3}\right)^{15} \) and subtract from 1: \[S_{15} = 2 \left(1 - 0.002646\right) = 2 \times 0.997354\]

Step 7: Final calculation

Finish the calculation: \[S_{15} \approx 2 \times 0.997354 = 1.994708\]

Key Concepts

geometric sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant non-zero number called the common ratio. This concept is essential in understanding various mathematical patterns and is widely used in fields such as finance, physics, and computer science.

In the given exercise, the sequence is \(\frac{2}{3}, \frac{2}{3} \times \frac{2}{3}, \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}, \text{...} \) , showing a clear geometric pattern.

Key characteristics of geometric sequences include:

• Each term is derived by multiplying the previous term by the common ratio.
• The sequence progresses either towards infinity or zero, depending on the common ratio.

sum formula

The sum formula for a geometric sequence allows us to quickly calculate the total of the first n terms. This formula is crucial because manually adding each term, especially for large sequences, can be tedious and error-prone. The general formula is:

\[ S_n = a_1 \frac{1-r^n}{1-r} \]

Where:

• \(a_1\) is the first term.
• \(r\) is the common ratio.
• \(n\) is the number of terms.

This formula helps by directly substituting the known values to find the sum. In our exercise, substituting the values into the formula simplifies the sum calculation process, helping us easily arrive at the sum of the first 15 terms.

common ratio

The common ratio is the factor by which we multiply each term in a geometric sequence to get the next term. It’s a critical component because it defines the sequence's growth or decay rate.

For a sequence \(a_1, a_2, a_3, \text{...} a_n\), the common ratio can be found using \( r = \frac{a_{n}}{a_{n-1}} \).
In the given problem, the common ratio is \( \frac{2}{3} \).

Characteristics of common ratio:

• If \(|r| < 1\), the terms get smaller and approach zero.
• If \(|r| > 1\), the terms grow larger.
• If \(r = -1\), the sequence alternates signs but stays constant in absolute value.

first term

The first term of a geometric sequence, denoted \(a_1\), is the initial value from which the sequence starts. It plays a crucial role in all subsequent calculations within the sequence. In simpler terms, you need to know where you start before you can find out where you're going.

In the provided example, the first term is \(\left(\frac{2}{3}\right)^1 = \frac{2}{3}\). This term is fundamental since it is the base value that is repeatedly multiplied by the common ratio to generate the entire sequence.

Remember, changing the first term changes the whole sequence, even if the common ratio remains constant. Thus, understanding and accurately identifying the first term is essential for solving problems involving geometric sequences.