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Chapter 14: Problem 16

Evaluate. See Example 1 $$ \sum_{i=1}^{4} 3^{i-1} $$

Short Answer

Expert verified
The sum is 40.

Step by step solution

01

Understanding the Problem

We need to evaluate the expression \( \sum_{i=1}^{4} 3^{i-1} \). This is a summation notation that tells us to sum up the values of \( 3^{i-1} \) from \( i = 1 \) to \( i = 4 \).
02

Calculating Each Term

Let's calculate each term individually:- For \( i = 1 \): \( 3^{1-1} = 3^0 = 1 \).- For \( i = 2 \): \( 3^{2-1} = 3^1 = 3 \).- For \( i = 3 \): \( 3^{3-1} = 3^2 = 9 \).- For \( i = 4 \): \( 3^{4-1} = 3^3 = 27 \).
03

Summing the Terms

Now, sum the calculated values from each term:\[ 1 + 3 + 9 + 27 \]
04

Calculating the Sum

Sum the numbers:\( 1 + 3 = 4 \)\( 4 + 9 = 13 \)\( 13 + 27 = 40 \)
05

Final Evaluation

The final summed value is \( 40 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are a type of mathematical function in which a constant base is raised to a variable exponent. In other words, the function is expressed as \(b^x\), where \(b\) is a fixed number known as the base, and \(x\) is the variable exponent. In the given exercise, we have an exponential function \(3^{i-1}\).There are several important properties to remember about exponential functions:
  • The base: Needs to be a positive number. In our problem, the base is \(3\).
  • The exponent: Directly affects the calculation. Here, the exponent is \(i-1\).
  • Exponential growth: As the exponent increases, the value of the function grows rapidly.
Understanding these properties can help efficiently solve problems involving exponential functions, as they quickly provide insight into how these functions behave.
Series Evaluation
Series evaluation involves finding the sum of a sequence of numbers or terms. In the context of our exercise, the series we are evaluating is the summation from \(i=1\) to \(i=4\) of \(3^{i-1}\).To evaluate such a series:
  • Identify each term of the series. In our case, these were \(3^0\), \(3^1\), \(3^2\), and \(3^3\).
  • Calculate each term individually before summing them together. Calculating these, we found the terms as 1, 3, 9, and 27.
  • Add up all the individual terms to get the series' total sum: \(1 + 3 + 9 + 27 = 40\).
Being methodical about each step in the process is crucial, as it allows for accurate calculation of the sum, providing a clear understanding of how each term contributes to the whole series.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations such as addition, subtraction, multiplication, and division. In our exercise, the expression \(3^{i-1}\) is an algebraic expression consisting of a constant base and a variable exponent achieved through the operation of exponentiation.When working with algebraic expressions:
  • Identify the components: Recognize constants (like the number 3 in \(3^{i-1}\)), variables (like \(i\)), and operations.
  • Manipulate the expressions systematically, if necessary, to perform calculations in a structured way. This includes careful handling of exponentiation and understanding its impact on the expression's value.
  • Simplify and solve expressions step-by-step to lessen errors and aid in understanding complex problems.
Algebraic expressions form the basis for many complex mathematical evaluations, and mastering the way to manipulate and understand them is pivotal in solving many types of mathematical problems.

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Most popular questions from this chapter

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