Problem 72 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{i=1}^{10}2\left(\dfrac{1}{4}\right)^{i - 1} \)

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is \( S_{10} = 2 \dfrac{1 - (\dfrac{1}{4})^{10}}{1 - (\dfrac{1}{4})} \)

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

The common ratio (r) is \( \dfrac{1}{4} \), the first term (a) is 2 and the number of terms (n) is 10.

Step 2: Use the formula for the sum of a finite geometric sequence

The formula to determine the sum of a finite sequence is \( S_n = a \dfrac{1 - r^n}{1 - r} \) where S is the sum of the sequence, n is the number of terms, a is the first term, and r is the common ratio.

Step 3: Substitute the values into the formula

Substitute the values of a, r, and n into the formula. \( S_{10} = 2 \dfrac{1 - (\dfrac{1}{4})^{10}}{1 - (\dfrac{1}{4})} \)

Step 4: Simplify the expression to get the final answer

Simplify the equation to calculate the final sum of geometric sequence. Keep in mind that you might need a calculator to solve it.

Key Concepts

Sum of Finite Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a constant called the common ratio. The sum of a finite geometric sequence involves adding up all these terms. For our exercise, the goal is to find the sum of these terms. This can be done using the formula for the sum of a finite geometric sequence, \[S_n = a \frac{1 - r^n}{1 - r}\]where:

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

This formula efficiently lets us calculate the sum without having to list and add each term manually. For example, in the given problem, you plug in the specific values of \(a = 2\), \(r = \frac{1}{4}\), and \(n = 10\). This makes calculating larger sequences manageable.

Common Ratio

The common ratio in a geometric sequence determines how each term relates to the one before it. It is found by dividing any term by the previous one in the sequence. If a sequence starts with a term \(a\) and has a common ratio \(r\), it will look like \(a, ar, ar^2, ar^3\), and so on. In our exercise, the common ratio is \(\frac{1}{4}\). This means each term is a quarter of the term before. Knowing the common ratio helps you understand whether the sequence is growing or shrinking, and by how much.

First Term

The first term of a geometric sequence is crucial since it serves as the starting point for calculating every subsequent term. In our example, the first term \(a\) is 2. From this point, by repeatedly applying the common ratio \(r\), you generate the rest of the sequence. This starting value can significantly impact the sum of the sequence, especially for sequences involving a large number of terms. For finite sequences, having the first term helps in accurately applying the formula to find the sum easily.

Number of Terms

The number of terms in a finite geometric sequence, denoted by \(n\), tells you how many terms in the series are being summed. In the given problem, there are 10 terms. This means that from the first term up to the 10th term, all values are part of the sum. The variable \(n\) is essential to the sum formula \[S_n = a \frac{1 - r^n}{1 - r}\]as it specifies how far along the geometric series you're calculating. The more terms there are, the greater potentially is the overall sum, which makes \(n\) a significant factor in these calculations.