Problem 35 - College Algebra

Question

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$\sum_{i=1}^{6}\left(\frac{1}{2}\right)^{i+1}$$

Step-by-Step Solution

Answer

\(\frac{63}{128}\)

Step 1 - Identify the first term (a) and the common ratio (r)

The first term (a) of our geometric sequence can be obtained by substituting the lower limit of the sum into the formula. It is \(a = (\frac{1}{2})^{1+1} = (\frac{1}{2})^2 = \frac{1}{4}\). The common ratio (r) is the base of the power, which here is \(\frac{1}{2}\).

Step 2 - Use the formula for the sum of a geometric sequence

The sum S of the first n terms of a geometric sequence can be expressed with the formula \(S = a \cdot \frac{1 - r^n}{1 - r}\). Substituting a, r and n = 6 into this formula, we have \(S = \frac{1}{4} \cdot \frac{1 - (\frac{1}{2})^6}{1 - \frac{1}{2}}\).

Step 3 - Calculate the sum

Now, perform the calculation: \(S = \frac{1}{4} \cdot \frac{1 - \frac{1}{64}}{\frac{1}{2}} = \frac{1}{4} \cdot \frac{63/64}{1/2} = \frac{63}{128}\).

Key Concepts

Sum of Geometric Series

When dealing with geometric sequences, finding the sum of several terms can be challenging. Luckily, the formula for the sum of the first \( n \) terms of a geometric sequence simplifies this task. This formula is expressed as:

• \( S = a \cdot \frac{1 - r^n}{1 - r} \)

Here, \( S \) represents the sum of the sequence, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms you want to sum up.
In the provided exercise, we used this formula to find the sum of the first 6 terms. We started by identifying the first term and common ratio. Then, we plugged these values into our formula.
Performing the step-by-step calculation allowed us to see how each component fits together. Always remember: use this formula anytime you need to quickly calculate the sum of terms in a geometric sequence.

Common Ratio

The common ratio in a geometric sequence is a crucial element. It denotes the factor by which each term is multiplied to get the next term. We can usually identify it by dividing any term by its preceding term.
In this exercise, the common ratio \( r \) is \( \frac{1}{2} \), evident from the expression \( (\frac{1}{2})^{i+1} \). This base of the exponential term represents how each subsequent term relates to the previous one.
Recognizing the common ratio is fundamental because it helps predict and find missing terms of the sequence. It's also a necessary component for calculating the sum of the series.

First Term of Sequence

Discovering the first term of a sequence involves substituting the index's starting value into the sequence's general term. For example, if we start the sum at \( i = 1 \), the first term \( a \) of our sequence is found by substituting \( i = 1 \) into \( (\frac{1}{2})^{i+1} \).

• First term: \( a = (\frac{1}{2})^2 = \frac{1}{4} \)

The first term is crucial as it initiates the sequence and serves as the starting point for calculating the sum of the sequence.
By knowing the first term, along with the common ratio, you can easily calculate specific terms and their sum using the geometric sequence formulas.