Problem 15 - Algebra 2 and Trigonometry

Question

In \(15-22 :\) a. Write each sum as a series. b. Find the sum of each series. $$ 3+\sum_{n=1}^{5} 3(2)^{n} $$

Step-by-Step Solution

Answer

The sum of the series is 189.

Step 1: Identify the Series

The given expression is \(3 + \sum_{n=1}^{5} 3(2)^n\). This means there is a constant 3 added to a geometric series where the general term is \(3(2)^n\).

Step 2: Write the Series

Write out the terms in the series from \(n=1\) to \(n=5\). Each term in the series is calculated as \(3(2)^n\):- For \(n=1\), the term is \(3(2)^1 = 6\),- For \(n=2\), the term is \(3(2)^2 = 12\),- For \(n=3\), the term is \(3(2)^3 = 24\),- For \(n=4\), the term is \(3(2)^4 = 48\),- For \(n=5\), the term is \(3(2)^5 = 96\).So, the series is \(3 + 6 + 12 + 24 + 48 + 96\).

Step 3: Calculate the Sum of the Series

Add the terms of the series together:- First, calculate the sum of the series terms except the initial 3: \(6 + 12 + 24 + 48 + 96 = 186\).- Then, add the initial 3 to the sum: \(3 + 186 = 189\).

Step 4: Conclusion

Therefore, the total sum of the series \(3 + \sum_{n=1}^{5} 3(2)^n\) is \(189\).

Key Concepts

Summation

Summation is a way of adding numbers together in a specific order. In mathematical terms, it involves adding a sequence of numbers, often represented as \( \sum \). This symbol, called "sigma," represents the act of summing a series of terms. In our problem, the expression \( \sum_{n=1}^{5} 3(2)^n \) indicates that we will add up each calculated term to get a total sum.
To understand it better, think of summation as a stack of dominoes, where each domino adds a certain value. You keep adding dominoes (or terms) until you reach the top of the stack, which gives you the final sum.
Summation is not just for numbers; it is a powerful tool used in various mathematical concepts, providing a manageable way to handle large amounts of data.

Series Addition

When we talk about series addition, we are referring to the process of summing a sequence of numbers or expressions systematically. In our exercise, the task is to write and then sum each term of a given series.
The series given by \( 3 + \sum_{n=1}^{5} 3(2)^n \) involves adding an initial constant, 3, to a geometric series. Each term in this series is found by multiplying a base number, 3, by powers of another number, 2, such as \( 3(2)^1, 3(2)^2, \) and so on.

• First, it involves identifying each term based on the given pattern.
• Then, the terms are written out and summed to find the total.

The key is to ensure each term is calculated correctly before adding them together to yield a precise total.

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." In our problem, the sequence involves terms like \( 3(2)^n \), where 2 is the common ratio.
To break it down:

• Start with the initial term, which is determined by multiplying the first term with the common ratio successively.
• Continue this multiplication until all required terms have been generated.

The critical aspect of a geometric sequence is the consistency of multiplication, which defines the pattern and allows us to predict future terms. Understanding geometric sequences is helpful as they appear in various real-life situations, from predicting population growth to calculating compound interest.

Algebra 2

Algebra 2 expands on the foundations laid in previous algebra courses, delving deeper into powerful mathematical concepts like sequences and series. In this context, we apply Algebra 2 techniques for a more advanced understanding of number sequences, such as a geometric series.
Here’s what involves:

• Manipulating expressions to recognize series patterns, like geometric sequences, and figuring out their sums.
• Utilizing algebraic rules to solve for unknowns or sum up sequences systematically.

In our particular equation, Algebra 2 helps us to methodically approach and solve the series problem: identifying each term, calculating their values, and finally summing them up. Mastering these techniques not only aids in solving mathematical problems but also sharpens analytical skills which are essential in tackling real-world challenges.