Problem 75 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{n=0}^{20}3\left(\dfrac{3}{2}\right)^n \)

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is \( 393727.08 \)

Step 1: Identifying the first number, common ratio, and the number of terms

The first term 'a' is 3, the common ratio 'r' is \( \frac{3}{2} \), and the number of terms 'n' is 21 (from n=0 to n=20).

Step 2: Substituting the values into the formula

Substitute these values into the formula to calculate the sum of the sequence. The sum \(S_n = 3 \cdot ((\frac{3}{2}) ^{21} - 1) / (\frac{3}{2} - 1)\)

Step 3: Simplifying the expression

Simplify the expression to find the final answer.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the series 2, 4, 8, 16, ..., the common ratio is 2 because each term is twice the previous term.

This pattern is particularly useful in various mathematical and real-world applications, such as calculating interest, analyzing growth patterns, and solving problems in physics and engineering.

Sum of Geometric Series

Finding the sum of a geometric series is a common problem in mathematics. For a finite geometric series, the sum can be calculated using a specific formula: \[ S_n = a \times \frac{r^n - 1}{r - 1} \]
where:

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

This formula applies only if the common ratio is not 1, since a ratio of 1 would result in a sequence where all terms are the same, and the formula would require division by zero.

Geometric Sequence Formula

A geometric sequence formula is used to find the nth term of a geometric series. The nth term is computed using the equation \[ a_n = a \times r^{(n-1)} \]
where \(a_n\) is the nth term, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

It's a powerful expression that encapsulates the behavior of the entire series at any arbitrary point, which in turn allows us to predict future terms or backtrack to find terms we may not have recorded.

Common Ratio

The common ratio in a geometric sequence is the factor by which successive terms are multiplied to obtain the next term. It is denoted by \(r\), and it's a key characteristic of geometric sequences. A positive common ratio results in a sequence that either increases (if \(r > 1\)) or decreases (if \(0 < r < 1\)) exponentially. When the common ratio is negative, the sequence will alternate between positive and negative values. It's crucial to identify the common ratio accurately as it not only defines the sequence but determines the convergence or divergence of its sum.