Problem 37 - A Graphical Approach to College Algebra

Question

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$\sum_{k=1}^{4}-2\left(\frac{1}{2}\right)^{k}$$

Step-by-Step Solution

Answer

The sum of the first five terms is \(-3.75\).

Step 1: Identify the Geometric Sequence Parameters

The formula for the sum of the first n terms of a geometric sequence is given by \( S_n = a \frac{1-r^n}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio. In this sequence, \( a = -2 \) and \( r = \frac{1}{2} \).

Step 2: Determine the Number of Terms

The problem asks for the sum of the first 4 terms of the sequence, so \( n = 4 \).

Step 3: Substitute Parameters into the Formula

Plug the values \( a = -2 \), \( r = \frac{1}{2} \), and \( n = 4 \) into the formula:\[ S_4 = -2 \cdot \frac{1 - \left(\frac{1}{2}\right)^4}{1 - \frac{1}{2}} \]

Step 4: Simplify the Expression

Calculate \( r^n = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \). Substitute this into the equation:\[ S_4 = -2 \cdot \frac{1 - \frac{1}{16}}{1 - \frac{1}{2}} = -2 \cdot \frac{\frac{15}{16}}{\frac{1}{2}} \]

Step 5: Final Calculation

Simplifying the expression:\[ S_4 = -2 \cdot \frac{15}{16} \cdot \frac{2}{1} = -2 \cdot \frac{30}{16} = -2 \cdot \frac{15}{8} = -2 \cdot 1.875 = -3.75 \]

Step 6: Rounding and Conclusion

Ensure the calculation is rounded as necessary. The final result \(-3.75\) is already rounded to the nearest hundredth.

Key Concepts

Sum of a Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. To find the sum of the first few terms, we use the sum formula for geometric sequences. This is given by \[S_n = a \frac{1 - r^n}{1 - r}\] where:

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

This formula provides a convenient way to add up terms in a geometric sequence without calculating each one individually. With this, you can quickly determine the total when a sequence either increases rapidly or decreases sharply. Remember to pay extra attention when the common ratio is less than one, as it significantly affects the sum.

Common Ratio

The common ratio in a geometric sequence is a constant that each term is multiplied by to get the next term. In the given exercise, the common ratio is \( r = \frac{1}{2} \). Using a common ratio less than one, each subsequent term in the sequence becomes smaller.In general, identifying the common ratio is key to understanding the pattern of the sequence. To find it, simply divide any term by the previous term. Here’s how:

• Take two consecutive terms from the sequence.
• Divide the latter by the former.
• For example, if your sequence was \(-2, -1, -0.5, -0.25,\ldots\), the ratio \(\frac{-1}{-2} = \frac{1}{2}\).

This ratio determines how quickly the sequence's terms grow or shrink, and plays a vital role in calculating the sum of terms.

First Term

The first term of a geometric sequence, often denoted as \( a \), is crucial in defining the sequence. It serves as the starting point. In the exercise, the first term is \(-2\.\) From here, you apply the common ratio to find the subsequent terms.This initial term is also significant for calculations involving the sum of terms. Since it multiplies the entire formula, its value directly influences the result. Whether it's larger or negative, you must account for the sign and magnitude when summing the sequence.Practically, the first term establishes the entire sequence's position and orientation, determining everything about the pattern from that point onward.

Number of Terms

The number of terms, represented by \( n \), indicates how many terms of the sequence you intend to sum. In this problem, you are summing the first 4 terms, which means \( n = 4 \).Knowing \( n \) is essential for using the formula. It affects the power to which the common ratio is raised in the expression \( r^n \). In this specific example, raising the common ratio \( \frac{1}{2} \) to the power of 4 results in \( \left( \frac{1}{2} \right)^4 = \frac{1}{16} \).An accurate count of terms ensures your sum reflects exactly what is intended, neither overshooting nor neglecting parts of the sequence. If you ever extend or limit the sequence differently, \( n \) changes accordingly, directly affecting both the calculations and the outcome.