Problem 35 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the finite geometric series. $$ 1+2+4+8+16+32+64+128 $$

Step-by-Step Solution

Answer

The sum of the series is 255.

Step 1: Identify the first term and the common ratio

The first step is to identify the first term \( a \) and the common ratio \( r \) of the geometric series. The first term \( a \) is 1 (as it is the first number in the series), and the common ratio \( r \) is 2 (since each term is multiplied by 2 to get the next term).

Step 2: Determine the number of terms \( n \)

The number of terms \( n \) in the given series can be counted directly. By counting each term, we see there are 8 terms: 1, 2, 4, 8, 16, 32, 64, and 128.

Step 3: Apply the formula for the sum of a finite geometric series

The formula for the sum \( S_n \) of the first \( n \) terms of a geometric series is given by:\[S_n = a \frac{r^n - 1}{r - 1} \]where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 4: Substitute the values into the formula

Substitute \( a = 1 \), \( r = 2 \), and \( n = 8 \) into the formula:\[S_8 = 1 \frac{2^8 - 1}{2 - 1} \]

Step 5: Calculate the sum

First, calculate \( 2^8 = 256 \). Then, \( 256 - 1 = 255 \). Finally, \( S_8 = 255 \), since we divide by \( 1 \). The sum of the finite geometric series is 255.

Key Concepts

Sum Formula for Finite Geometric Series

The sum of a finite geometric series can be efficiently calculated using a specific formula. This formula helps you find the total sum of all terms in the series without having to add each one manually.
This is especially useful when dealing with large numbers of terms. The sum formula for a geometric series is represented as:

• \(S_n = a \frac{r^n - 1}{r - 1} \)

Here:

• \(a\) is the first term of the series.
• \(r\) is the common ratio.
• \(n\) is the number of terms in the series.

This formula subtracts 1 from \(r^n\) because it adjusts for the starting point of the series. Dividing by \(r - 1\) simplifies the sum, as you're accounting for how the series progresses at a constant multiplier.

Understanding the Common Ratio

The common ratio \(r\) is a crucial component of a geometric series. It is the factor by which each term in the series is multiplied to obtain the next term.
This ratio remains constant throughout the series and distinguishes a geometric series from other types of series. To determine the common ratio, divide any term in the series by the preceding term.

• Example: In the series 1, 2, 4, 8, ..., the common ratio \(r\) is \(\frac{2}{1} = 2\).

The ability to find and confirm the common ratio ensures that the sequence is indeed geometric. It also plays a key role in using the sum formula correctly, as it is required to calculate the power of \(r\) raised to the number of terms \(n\).

Determining the Number of Terms

The number of terms \(n\) in a finite geometric series is simply the count of individual terms present from start to finish. Knowing \(n\) is essential to properly applying the sum formula.
In the context of a finite series, this quantifies how many times the sequence progresses under the common ratio. To determine the number of terms:

• Simply count each number in the sequence.
• For example, the series 1, 2, 4, 8, 16, 32, 64, 128 contains 8 terms.

Each term corresponds to a specific step in applying the ratio, which is why taking a careful count is important when solving for the sum.

Identifying the First Term

The first term, denoted as \(a\), sets the foundation for the entire geometric series. It is the initial value from which all other terms are progressively calculated by applying the common ratio. To find the first term:

• Look at the start of the sequence. This is usually straightforward.
• For example, in the series 1, 2, 4, 8, ..., the first term \(a\) is 1.

Understanding the role of the first term is vital as it impacts the sum calculation; you’re essentially building the sequence upon this primary value.