Problem 17 - Algebra 2 and Trigonometry

Question

In \(15-22 :\) a. Write each sum as a series. b. Find the sum of each series. $$ 10+\sum_{n=1}^{5} 10\left(\frac{1}{2}\right)^{n} $$

Step-by-Step Solution

Answer

The series sum is 19.6875.

Step 1: Understanding the Series

The given expression is a sum of a number 10 and a geometric series represented by \( \sum_{n=1}^{5} 10 \left( \frac{1}{2} \right)^{n} \). This means we first need to write the series explicitly and then find its sum.

Step 2: Writing the Series

Expand the series part: \( \sum_{n=1}^{5} 10 \left( \frac{1}{2} \right)^{n} = 10 \times \left( \frac{1}{2}^1 \right) + 10 \times \left( \frac{1}{2}^2 \right) + 10 \times \left( \frac{1}{2}^3 \right) + 10 \times \left( \frac{1}{2}^4 \right) + 10 \times \left( \frac{1}{2}^5 \right) \).

Step 3: Calculating Each Term of the Series

Calculate each term: - First term: \( 10 \times \frac{1}{2} = 5 \)- Second term: \( 10 \times \left( \frac{1}{2} \right)^2 = 2.5 \)- Third term: \( 10 \times \left( \frac{1}{2} \right)^3 = 1.25 \)- Fourth term: \( 10 \times \left( \frac{1}{2} \right)^4 = 0.625 \)- Fifth term: \( 10 \times \left( \frac{1}{2} \right)^5 = 0.3125 \)

Step 4: Writing the Complete Sum as a Series

Include the initial number 10 and list all calculated terms: \( 10 + \sum_{n=1}^{5} 10 \left( \frac{1}{2} \right)^{n} = 10 + 5 + 2.5 + 1.25 + 0.625 + 0.3125 \)

Step 5: Calculating the Total Sum of the Series

Add all terms together:\( 10 + 5 = 15 \)\( 15 + 2.5 = 17.5 \)\( 17.5 + 1.25 = 18.75 \)\( 18.75 + 0.625 = 19.375 \)\( 19.375 + 0.3125 = 19.6875 \)

Step 6: Final Solution

The entire expression simplifies to the series sum of \( 19.6875 \).

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the "common ratio." This pattern makes it easy to identify and predict the sequence. For example, in the problem provided, we have a sequence where each term is produced by multiplying the previous term by \( \frac{1}{2} \). The first term of the sequence is \( 10 \times \frac{1}{2} = 5 \), indicating that 5 is the result of the common ratio applied to 10.
Geometric sequences are essential in mathematics because they allow for easy manipulation and calculation through their regularity and predictability. Understanding this concept is key to recognizing how the terms in a sequence relate to each other through multiplication.

Summation

Summation is the operation of adding a sequence of numbers, typically expressed with the summation notation \( \sum \). In the provided exercise, we use summation to add the terms of the geometric series generated by the formula \( \sum_{n=1}^{5} 10 \left( \frac{1}{2} \right)^{n} \). Each 'n' value represents a consecutive term in the sequence.
To execute the summation, we compute each term in the sequence separately, as shown in the exercise, and then sum all computed terms together, starting from the initial number, 10. Understanding summation helps us efficiently find the total of complex mathematical series via straightforward arithmetic.

Mathematical Series

A mathematical series is the sum of the terms of a sequence. When you take the sum of a geometric sequence, it becomes a geometric series. In simpler words, each term of the sequence adds up to form the series, and this cumulative sum is what we seek in the exercise
In our specific case, the series includes both the single number 10 and the sum of the products from the geometric sequence terms: \( 10 + 5 + 2.5 + 1.25 + 0.625 + 0.3125 \). Once all the terms are identified, they are added to find the sum of the series: \( 19.6875 \). Understanding the concept of series helps in comprehending not just a single element but how they cooperate cumulatively within a given mathematical problem.

Algebra 2

In Algebra 2, we delve into more complex mathematical concepts, including sequences and series, which form critical components. Understanding geometric sequences and series is often a part of the Algebra 2 curriculum. These topics help build the foundation for higher-level math courses.
Algebra 2 not only involves working with numerical operations and equations but also enhancing one's analytical thinking skills. The geometric series in this exercise helps explain how variables and constants collaborate systematically to form sequences and series that have real-world applications, such as in financial calculations or predictive modeling.