Problem 76 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{n=0}^{40}5\left(\dfrac{3}{5}\right)^n \)

Step-by-Step Solution

Answer

The sum of the given finite geometric series is found by substituting the given values into the formula for the sum of a finite geometric sequence.

Step 1: Identify the first term (a), the common ratio (r) and the number of terms (n)

Here, the first term \(a\) is 5, the common ratio \(r\) is \(\frac{3}{5}\) and the number of terms \(n\) is \(40 + 1 = 41\) (since the series starts from 0).

Step 2: Apply the formula for the sum of a finite geometric series

The formula for the sum \(S\) of a finite geometric series is given by \(S = a \frac{1 - r^n}{1 - r}\). Let's fill our figures into the formula.

Step 3: Substitute the values into the formula and solve

\(S = 5 \frac{1 - \left(\frac{3}{5}\right)^{41}}{1 - \frac{3}{5}}\). By simplifying the above equation we can calculate the sum of the finite geometric sequence.

Key Concepts

sum of geometric series

The sum of a geometric series involves finding the total of all terms in a geometric sequence. A geometric series is a series where each term is derived by multiplying the previous one by a constant factor known as the common ratio. In the context provided, we are dealing with a *finite* geometric series, meaning that the series has a definite number of terms.

To find the sum, we use a specific formula, which simplifies the process significantly. The formula is: \[ S = a \frac{1 - r^n}{1 - r} \]where:

• \( S \) is the sum of the series,
• \( a \) is the first term in the series,
• \( r \) is the common ratio, and
• \( n \) is the number of terms.

In the solution, using this formula helped calculate the total sum of the sequence by substituting the given values and simplifying the equation. The formula ensures that as long as you have the first term, common ratio, and number of terms, solving for the sum is straightforward.

This method is particularly useful in problems involving large sequences where calculating each term and then summing them would be cumbersome.

geometric sequence formula

A geometric sequence is a list of numbers where each term after the first is determined by multiplying the previous term by a constant, known as the common ratio.

The general formula for the nth term of a geometric sequence is given by:\[ a_n = a \cdot r^{(n-1)} \]where:

• \( a_n \) is the nth term,
• \( a \) is the first term,
• \( r \) is the common ratio,
• \( n \) is the term number.

Using this formula helps determine any term in the sequence without listing out all previous terms, making it quite efficient and less error-prone.

In practical applications, knowing how to derive terms in a sequence quickly is essential, such as calculating interest in financial problems or analyzing growth patterns in populations where multiplication by a constant rate is involved.

common ratio

The common ratio is a critical component in understanding geometric sequences and series. It is the factor by which we multiply each term to get the next term in the series. Being a constant, this ratio provides the regularity and pattern that defines geometric sequences.

For instance, in our exercise, the common ratio is \( \frac{3}{5} \),meaning each term is \( \frac{3}{5} \)of the previous term. If the common ratio is greater than 1, the terms will grow larger, while a common ratio between 0 and 1 will cause the terms to diminish. Negative common ratios can cause the terms to alternate in sign.

Understanding the common ratio can also help predict the behavior of the series:

• {large or smaller terms versus the initial term, which direction they head in (growing or shrinking), and even how quickly they might approach a limit.
• The ratio dictates the growth or decay rate, making it invaluable in both theoretical explorations and practical applications like calculating depreciation or growth.