Problem 85 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{i=1}^{10}5\left(-\dfrac{1}{3}\right)^{i - 1} \)

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is approximately 3447.44.

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

Here in the given exercise, the first term is a = 5, the common ratio is r = -1/3, and the number of terms (n) = 10.

Step 2: Apply the formula for sum of finite geometric sequence

The formula for sum of finite geometric sequence \(S_n = a \dfrac{1- r^n}{1 - r}\). Here a = 5, r = -1/3 and n = 10. By substituting the values we get, \(S_{10} = 5 \dfrac{1 - (-1/3)^{10}}{1 + 1/3}\)

Step 3: Compute the sum

Computing the values we get, \(S_{10} = 5 \dfrac{1 - (-1/3^{10})}{1 + 1/3} = 5 \dfrac{1 - (1/59049)}{4/3} = 5 \dfrac{(59049 - 1)/59049}{4/3} = 5 * 14714/59049 * 3/4 = 3447.44\). So, the sum of the finite geometric series is approximately 3447.44.

Key Concepts

Geometric Sequence Formula

A geometric sequence is a series 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 geometric sequence formula is essential for identifying terms in such a sequence. If the first term is denoted by \( a \) and the common ratio by \( r \), the general formula for the \( n \)-th term \( a_n \) is given by:\[ a_n = a \cdot r^{(n-1)} \]This formula helps you find any term in the sequence without listing all of the preceding terms. For example, in a sequence where \( a = 5 \) and \( r = -\frac{1}{3} \), you could easily find the 10th term by substituting these values into the formula.

Sum of Geometric Series

The sum of a geometric series, particularly useful in finite sequences, is the total of all terms added together. The formula to calculate the sum \( S_n \) of the first \( n \) terms of a geometric sequence with initial term \( a \) and common ratio \( r \) is:\[ S_n = a \cdot \frac{1- r^n}{1 - r} \]This formula simplifies the process of finding the total sum without individually adding each term. In the exercise, this formula is utilized with \( a = 5 \), \( r = -\frac{1}{3} \), and \( n = 10 \). Plugging these values into the formula gives:\[ S_{10} = 5 \cdot \frac{1 - (-\frac{1}{3})^{10}}{1 + \frac{1}{3}} \]This yields an efficient calculation process, ultimately showing that the sum of the sequence is approximately 3447.44.

Common Ratio in Sequences

The common ratio is a key element in geometric sequences. It is the constant factor between consecutive terms. The value of the common ratio \( r \) determines the nature of the sequence:

• If \( 0 < r < 1 \), the terms will decrease and approach zero.
• If \( r = 1 \), all terms are the same.
• If \( r > 1 \), each subsequent term grows larger.
• If \( r < 0 \), the terms alternate in sign.

In the provided exercise, the common ratio is \( r = -\frac{1}{3} \), indicating an alternating sequence where each term is one-third the size of the preceding term before sign change. Understanding \( r \) is crucial for correctly applying the geometric sequence and series formulas.