Problem 27 - A Graphical Approach to College Algebra

Question

Use a formula to find each sum. $$\sum_{i=1}^{5} 3^{i}$$

Step-by-Step Solution

Answer

The sum of the series is 363.

Step 1: Understand the Problem

The task is to calculate the sum of the powers of 3 from 1 up to 5. In other words, we need to find \(3^1 + 3^2 + 3^3 + 3^4 + 3^5\).

Step 2: Identify the Formula

This is a geometric series where the first term \(a = 3\) and the common ratio \(r = 3\). The formula for the sum \(S_n\) of the first n terms of a geometric series is \(S_n = a \frac{r^n - 1}{r - 1}\).

Step 3: Apply the Formula

Substitute the values into the formula: \(a = 3\), \(r = 3\), and \(n = 5\). Thus, \(S_5 = 3 \frac{3^5 - 1}{3 - 1}\).

Step 4: Calculate Powers and Differences

First, calculate \(3^5\): \[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \]Substitute back into the formula: \[ S_5 = 3 \frac{243 - 1}{2} \]

Step 5: Simplify and Solve

Now simplify: \[ S_5 = 3 \frac{242}{2} = 3 \times 121 = 363 \] Therefore, the sum is 363.

Key Concepts

Powers of Numbers

Understanding powers of numbers is essential in mathematics, especially when dealing with series. A power refers to the operation of raising a base number to an exponent. For instance, in the expression \(3^i\), 3 is the base, and \(i\) is the exponent or power. The value of \(3^i\) means multiplying 3 by itself \(i\) times. For example:

• \(3^1 = 3\)
• \(3^2 = 9\)
• \(3^3 = 27\)
• \(3^4 = 81\)
• \(3^5 = 243\)

Each step increases the power, illustrating how exponential growth works. Recognizing and calculating powers is vital for understanding the behavior of geometric sequences, where each term is a power of a specific number.

Sum of Series

A series is the sum of a sequence of terms. When dealing with a series, we often find ourselves looking for a quick way to calculate the sum of its elements rather than adding them one by one. This situation arises frequently in geometric series, which are characterized by each term being a constant multiple of the preceding term.

In our example, we found the sum of the series \(3^1 + 3^2 + 3^3 + 3^4 + 3^5\). By adding these terms, we could simply find the total, but that would be tedious for longer sequences. Therefore, understanding and applying the geometric series sum formula can save time and effort.

• Identify the first term (\(a\)) and the common ratio (\(r\)).
• Use the formula for the sum of the first \(n\) terms.

Being able to efficiently sum series is a key mathematical skill, particularly in more complex algebraic or calculus problems.

Geometric Progression Formula

The geometric progression formula is a powerful tool that lets us sum the terms of a geometric series efficiently. A geometric progression is defined by its first term \(a\), its common ratio \(r\), and the number of terms \(n\).
The formula for the sum \(S_n\) of the first \(n\) terms of a geometric series is:\[S_n = a \frac{r^n - 1}{r - 1}\]This formula allows us to calculate the sum without individually adding each term. Instead, we plug in our known values: the first term, common ratio, and number of terms. In the exercise, we had:

• First term \(a = 3\)
• Common ratio \(r = 3\)
• Number of terms \(n = 5\)

Substituting these into the formula gives us:\[S_5 = 3 \frac{3^5 - 1}{3 - 1} = 3 \frac{243 - 1}{2} = 3 \times 121 = 363\]This formula simplifies the process, even for a longer series, ensuring the calculation is both quick and accurate.