Problem 95 - Precalculus with Limits

Question

In Exercises 85-96, find the sum. \( \displaystyle \sum_{i=1}^{4} 2^i \)

Step-by-Step Solution

Answer

The sum of the geometric series is 30.

Step 1: Identify the values

Identify the first term (\(a\)), the common ratio (\(r\)), and the number of terms (\(n\)). In this case, the first term is \(2^1 = 2\), the common ratio is also 2 (since each subsequent term is multiplied by 2), and there are 4 terms since the sum is from \(i=1\) to \(i=4\).

Step 2: Use the sum formula for a geometric series

The formula for the sum (\(S\)) of the first \(n\) terms of a geometric sequence where \(a\) is the first term and \(r\) is the common ratio is \(S = a \times \frac{1 - r^n}{1 - r}\). Plug in the identified values into this formula.

Step 3: Calculate the Sum

Substitute \(a = 2\), \(r = 2\), and \(n = 4\) into the formula: \(S = 2 \times \frac{1-2^4}{1-2} = 2 \times \frac{-15}{-1}= 30\)

Key Concepts

Sum of a Geometric Series

Understanding how to find the sum of a geometric series is an invaluable skill in many aspects of mathematics. A geometric series is the sum of a sequence of terms, each of which is multiplied by a constant factor from the previous term. The challenge often lies not just in performing calculations, but in recognizing the structure of a geometric series.

In the presented exercise, we deal with the geometric series \( \sum_{i=1}^{4} 2^i \), which can be explicitly written out as \(2^1 + 2^2 + 2^3 + 2^4\). The common ratio here is 2, as each term is doubled to get the next. To find the series' sum, we utilize a simple formula: \(S = a \times \frac{1 - r^n}{1 - r}\) where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. When these values are substituted, carrying out the arithmetic leads to the desired sum.

Geometric Sequence

A geometric sequence is a string of numbers where each successive term is derived by multiplying the previous one by a fixed, non-zero number called the common ratio. The structure of a geometric sequence is central to various mathematical disciplines and real-world applications like calculating interest rates or population growth.

Our example features the geometric sequence \(2, 4, 8, 16\), which follows from the general term \(2^i\). Each term can be obtained by multiplying the preceding term by our common ratio of 2. This kind of sequence, where the operation between terms is multiplication, differs importantly from an arithmetic sequence, where the operation is addition.

Series Convergence

Determining if a series converges is key to understanding its behavior and applying this to real-world contexts. A geometric series converges if its common ratio \(r\) has an absolute value less than 1, which means the terms get progressively smaller and the series approaches a finite value. When \(|r| >= 1\), however, the series does not converge and instead grows indefinitely.

In our example, the common ratio is 2, leading to a series that doesn’t converge, as it's greater than 1. It's important to note this concept applies to infinite series, whereas the example provided is a finite sum, hence convergence doesn't directly influence the outcome.

Arithmetic Operations with Exponents

Operations with exponents form the bedrock of working with geometric series. The exponents indicate the number of times the base number is used as a factor. Arithmetic operations with exponents must follow specific rules, such as when multiplying powers with the same base, you add the exponents, and when raising a power to a power, you multiply the exponents.

In the case of our exercise, calculating \( 2^i \) for \( i = 1 \) to \( 4 \) involves recognizing these rules. Utilizing properties of exponents simplifies the process of finding the terms of the sequence and aids in calculating the sum. Mastery of these rules enables a clear path to evaluate operations involving exponential expressions.