Problem 6 - Algebra 2

Question

Find the sum of each geometric series. $$ \sum_{n=1}^{5} \frac{1}{4} \cdot 2^{n-1} $$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{31}{4} \) or 7.75.

Step 1: Identify the Series Parameters

The given series is \( \sum_{n=1}^{5} \frac{1}{4} \cdot 2^{n-1} \). This is a geometric series because it involves a constant ratio between successive terms. The first term \( a \) is found by setting \( n = 1 \):\( \frac{1}{4} \cdot 2^{1-1} = \frac{1}{4} \cdot 2^0 = \frac{1}{4} \). The common ratio \( r \) is \( 2 \), since each term is multiplied by \( 2 \) compared to the previous term.

Step 2: Apply the Formula for the Sum of a Geometric Series

The formula for the sum \( S \) of the first \( n \) terms of a geometric series is \( S_n = a \frac{1-r^n}{1-r} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. Here, \( a = \frac{1}{4}, r = 2, \) and \( n = 5 \). Substitute these values into the formula: \[ S_5 = \frac{1}{4} \frac{1-2^5}{1-2} \].

Step 3: Calculate Powers and Simplify the Expression

Calculate \( 2^5 \) which is \( 32 \). Substitute this value into the formula: \[ S_5 = \frac{1}{4} \frac{1-32}{-1} = \frac{1}{4} \cdot 31 \].

Step 4: Perform the Final Multiplication

Multiply \( \frac{1}{4} \) by \( 31 \) to find the sum of the series: \( S_5 = \frac{31}{4} \). Convert this fraction to a decimal if necessary: \( \frac{31}{4} = 7.75 \).

Key Concepts

Sum of a Geometric Series

A geometric series is a sum of terms where each term is obtained by multiplying the previous term by a constant known as the common ratio. Finding the sum of a finite geometric series is straightforward when you use the right formula. The formula for the sum of the first \( n \) terms of a geometric series is:

• \( S_n = a \frac{1-r^n}{1-r} \)

where:

• \( S_n \) is the sum of the first \( n \) terms,
• \( a \) is the first term,
• \( r \) is the common ratio, and
• \( n \) is the total number of terms.

To solve the exercise provided, we identified the first term \( a = \frac{1}{4} \), the common ratio \( r = 2 \), and the number of terms \( n = 5 \). By substituting these values into the formula, we calculated the sum of the series as \( S_5 = \frac{31}{4} \) or converted into a decimal, 7.75.
Using the sum formula efficiently helps you find the sum of terms quickly without calculating each term individually.

Common Ratio

The common ratio is a key element in a geometric sequence or series. It determines how you get from one term to the next. You can find it by dividing any term in the sequence by the previous term:

• \( r = \frac{a_{n}}{a_{n-1}} \)

In our given exercise, each term was formed by multiplying the previous term by \( 2 \). Hence, the common ratio \( r \) is \( 2 \).
This value illustrates how the series progresses. In this case, it shows a doubling pattern. Understanding the common ratio helps predict future terms and is essential when using the formula for the sum of a geometric series. The common ratio must be consistent for the series to qualify as geometric.

First Term

In any geometric series, the first term is the starting point for the sequence of numbers. The first term, often represented as \( a \), is critical for calculating both individual terms and the sum of the series.In the exercise, the first term \( a \) for the series \( \sum_{n=1}^{5} \frac{1}{4} \cdot 2^{n-1} \) was calculated by setting \( n = 1 \). This gives:

• \( a = \frac{1}{4} \cdot 2^{1-1} = \frac{1}{4} \)

This beginning term is pivotal in determining the overall makeup of the series. Understanding and identifying the first term is necessary when solving problems involving geometric sequences or series. Knowing the first term allows you to apply relevant formulas to find sums or predict future terms efficiently.