Problem 54 - Precalculus: Mathematics for Calculus

Question

Find the sum. $$\sum_{i=1}^{3} i 2^{i}$$

Step-by-Step Solution

Answer

34

Step 1: Understanding the Expression

The given expression is a summation: \( \sum_{i=1}^{3} i \cdot 2^{i} \). This means we need to compute the sum of \( i \cdot 2^{i} \) for \( i \) values starting at 1 and ending at 3.

Step 2: Compute Individual Terms

Calculate each term in the series:- When \( i=1 \), the term is \( 1 \cdot 2^{1} = 2 \).- When \( i=2 \), the term is \( 2 \cdot 2^{2} = 8 \).- When \( i=3 \), the term is \( 3 \cdot 2^{3} = 24 \).

Step 3: Sum the Terms

Add the calculated terms together: \\( 2 + 8 + 24 = 34 \).

Step 4: Verify the Calculation

Double-check each operation to ensure accuracy:- Each exponential operation is valid since \( 2^{1} = 2 \), \( 2^{2} = 4 \), and \( 2^{3} = 8 \).- Each multiplication was done correctly. Therefore, the sum \( 2 + 8 + 24 = 34 \) is accurate.

Key Concepts

Understanding Series in Mathematics

In mathematics, a series is essentially the sum of the terms of a sequence. When dealing with series, we often use the summation notation, represented by the Greek letter sigma (\( \Sigma \)), to denote the addition of a series of numbers.
For example, in our original exercise, the expression \( \sum_{i=1}^{3} i \cdot 2^{i} \) is a series. Here, the variable \( i \) takes on integer values starting from 1 up to 3. Each term in this series is formed by multiplying the index \( i \) by \( 2^{i} \), and the task is to sum these terms together.
Here's how a series works:

• The sequence of numbers forms the basis of the series, each number is referred to as a term.
• The terms are added together to form the series.
• The starting and ending points of the series are defined by the limits of the summation notation.

Understanding series is crucial for solving problems that involve repeated addition of terms, especially in calculus and algebra.

Exploring Exponential Functions

Exponential functions are functions where variables appear in the exponent, usually in the form \( a^{x} \), where \( a \) is a constant base and \( x \) is the exponent. These functions display rapid growth or decay depending on the base value.
In our exercise, the exponential function is represented by \( 2^{i} \). Here, 2 is the base, and \( i \) is the variable exponent. As \( i \) increases, the value of \( 2^{i} \) increases, showcasing the nature of exponential growth.
Important properties of exponential functions include:

• Rapid Growth: As the exponent increases, the function value grows quickly.
• Continuous Growth: Exponential functions model real-world scenarios like population growth and compound interest.
• Basic Operations: Base can be any positive number, but in most mathematical contexts, a base greater than 1 represents growth.

Mastering exponential functions is vital for understanding complex mathematical concepts, as many equations and real-life applications rely on them to depict dynamic changes.

Performing Arithmetic Operations in Series

Arithmetic operations form the foundation for calculating sums in series. The operations include addition, multiplication, and sometimes subtraction and division.
Calculating the sum of a series involves performing these operations sequentially and carefully. For the given exercise, the steps were:

• Multiplication: Each term of the series \( i \cdot 2^{i} \) demands multiplying \( i \) (the position of the term) by \( 2^{i} \) (the exponential component).
• Addition: The calculated terms are then added together to get the final sum. In the solution, terms 2, 8, and 24 were added.
• Verification: Once all operations are completed, it's crucial to verify every step to ensure accuracy, like checking if \( 2 + 8 + 24 = 34 \).

By practicing these arithmetic operations, students improve their problem-solving abilities and become proficient at handling numbers in both simple and complex series calculations.