Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Find the sum for each series. $$\sum_{i=-1}^{2} 5(2)^{i}$$

Step-by-Step Solution

Answer

The sum of the series is 37.5.

Step 1: Understand the series

The given series is a summation notation that can be expanded as follows: \( \sum_{i=-1}^{2} 5(2)^{i} = 5(2)^{-1} + 5(2)^{0} + 5(2)^{1} + 5(2)^{2} \). Our goal is to evaluate this expression by calculating each term and then summing them up.

Step 2: Calculate each term

Let's calculate each term one by one: 1. \( 5(2)^{-1} = 5 \times \frac{1}{2} = 2.5 \)2. \( 5(2)^{0} = 5 \times 1 = 5 \)3. \( 5(2)^{1} = 5 \times 2 = 10 \)4. \( 5(2)^{2} = 5 \times 4 = 20 \)

Step 3: Sum the calculated terms

Add up the calculated terms: \( 2.5 + 5 + 10 + 20 = 37.5 \).Thus, the sum of the series is 37.5.

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the "common difference."
For example, in the sequence 2, 4, 6, 8, the common difference is 2, as each term increases by 2 from the one before.

• Arithmetic series are represented by the formula: \( S_n = \frac{n}{2} (a + l) \), where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, and \( l \) is the last term.
• They can also be represented as: \( S_n = \frac{n}{2} [2a + (n-1)d] \), where \( d \) is the common difference.

Although the original exercise isn't directly related to an arithmetic series because the sequence grows exponentially rather than linearly, understanding arithmetic series helps provide context for different types of series, including more complex ones like geometric and exponential. Knowing how to manipulate and find sums in arithmetic series can be very helpful when dealing with more complex summation problems.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are represented in the form \( f(x) = a \cdot b^x \), where \( a \) is a constant, \( b \) is the base, and \( x \) is the exponent.

• The base \( b \) must be a positive real number, and it should not equal 1.
• Exponential functions can grow rapidly, which makes them handy in modeling scenarios like population growth and compound interest.
• Understanding exponential growth is essential as they appear frequently in sequences and series like the one in the exercise.

In the given exercise, \( 5(2)^{i} \) is an example of an exponential function, where \( 5 \) is a constant multiplier, \( 2 \) is the base, and \( i \) is the exponent, which can change values. Calculating the sum of terms where the base is raised to different powers (like \( i = -1, 0, 1, 2 \)) is a common step in problems involving exponential series.

Precalculus

Precalculus is an essential course that prepares students for calculus by establishing a strong mathematical foundation. It includes the study of functions and limits, trigonometry, and sequences and series.
Among other topics, precalculus introduces critical concepts like summation notation and series, which are crucial for understanding calculus.

• Precalculus emphasizes the understanding of different functions including polynomial, rational, exponential, and logarithmic functions.
• It helps students develop the ability to manipulate algebraic expressions and delve into more complex mathematical problems.
• By learning about sequences and series, students can start exploring the relationships between different quantities.

The exercise provided involves sequences in exponential form, and the summation notation \( \sum \) is a key concept in precalculus. Understanding how to resolve these notations is fundamental in transitioning to higher-level math. Mastery of precalculus allows students to approach problems like the one in the exercise with confidence, setting the stage for tackling calculus topics.