Problem 31 - 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. $$\sum_{k=4}^{10}(-2)^{k}$$

Step-by-Step Solution

Answer

The sum of the sequence is 688.

Step 1: Identify the Sequence

The given sequence involves powers of -2, starting from the 4th power up to the 10th power. The general term for the sequence can be expressed as \((-2)^k\), where \(k\) ranges from 4 to 10.

Step 2: List the Terms of the Sequence

List each term of the sequence from \(k=4\) to \(k=10\):- \((-2)^4 = 16\)- \((-2)^5 = -32\)- \((-2)^6 = 64\)- \((-2)^7 = -128\)- \((-2)^8 = 256\)- \((-2)^9 = -512\)- \((-2)^{10} = 1024\)

Step 3: Add the Terms Together

Sum all of the terms calculated previously:\[16 + (-32) + 64 + (-128) + 256 + (-512) + 1024\].

Step 4: Calculate the Sum

Calculate the sum of the series:\[16 + (-32) + 64 + (-128) + 256 + (-512) + 1024 = 688\].

Key Concepts

Sum of a Geometric Sequence

When dealing with geometric sequences, one of the key areas of interest is finding the sum of the terms, known as the "sum of a geometric sequence." This is particularly useful when the sequence is large. Remember, a geometric sequence is characterized by each term being a constant multiple, called the common ratio, of the previous one. In our example, however, instead of finding a common ratio, we directly focused on calculating the sum from known terms, that are powers of

• -2 to various powers, namely from 4 to 10.
• This specific sequence uses \[(a_1, a_2, ..., a_n) = ((-2)^4, (-2)^5, ..., (-2)^{10})\]
• where each term is raised to a consecutive power starting from 4 up to 10.

Thus, to find the sum \(S(n)\), all terms from \((k=4)\) to \((k=10)\) were listed, and then summed up sequentially.

Series Calculation

In calculating the series, the process involves explicitly expressing each term of the given sequence before adding them together. This sequence calculation can be simplified into a series of steps:

• Identify the Pattern: Here, we see that each term is the result of raising -2 to a consecutive power starting at 4.
• List and Calculate Each Term: As detailed in the solution, every term from \((-2)^4\) to \((-2)^{10}\) was individually calculated.
• Add the Terms: Finally, these terms were summed to derive a total value for the series.

This methodology allows for an accurate addition of sequences where terms vary, ensuring each calculation contributes to a collective sum. It is a straightforward approach, ensuring each step aligns with mathematical order and correctness.

Powers of Numbers

Understanding powers of numbers is crucial when working with sequences like the one in the exercise. A power of a number is the result of multiplying that number by itself a certain number of times. In this system of notation,

• the base number is -2.
• The exponent indicates how many times -2 is used as a factor.

For example, \((-2)^4\) would be understood as \(-2 \times -2 \times -2 \times -2\). When calculating each power in the sequence, alternating signs occur since

• an even power results in a positive number.
• An odd power results in a negative number.

This pattern is significant for calculating sums as each subsequent power sways the sum up and down due to positive and negative sign alternation. Mastery over understanding powers ensures accurate calculation in more complex algebraic scenarios.