Problem 11 - Calculus and its applications

Question

Express \(\sum_{i=1}^{4} 2^{i}\) without using summation notation.

Step-by-Step Solution

Answer

The sum is 30.

Step 1: Understand the Summation

The expression \(\sum_{i=1}^{4} 2^{i}\) represents the sum of the terms \(2^{1}\), \(2^{2}\), \(2^{3}\), and \(2^{4}\). We need to calculate each term individually and then add them together.

Step 2: Calculate Each Term in the Series

First, calculate \(2^{1} = 2\), then \(2^{2} = 4\), next \(2^{3} = 8\), and finally \(2^{4} = 16\). These are the powers of 2 from the first to the fourth term.

Step 3: Add the Terms Together

Now add the results from Step 2 together: \[2 + 4 + 8 + 16.\]This will give you the total sum of the expression.

Step 4: Simplify the Sum

Add the terms:\[2 + 4 + 8 + 16 = 30.\]Thus, \(\sum_{i=1}^{4} 2^{i}\) simplifies to 30.

Key Concepts

Powers of integers

When you hear 'powers of integers,' it refers to the operation of raising a number to a certain exponent. Think about it as multiplying a number by itself a number of times. For example, in the expression \(2^{i}\), \(2\) is called the base and \(i\) is the exponent. This means multiplying \(2\) by itself \(i\) times.

• \(2^1 = 2\): This means \(2\) multiplied by itself 0 more times (it is simply \(2\)).
• \(2^2 = 4\): Here, it is \(2 \times 2\).
• \(2^3 = 8\): This means \(2 \times 2 \times 2\).
• \(2^4 = 16\): This means \(2 \times 2 \times 2 \times 2\).

Understanding how to calculate powers of integers is crucial when dealing with sequences and series, especially in summation notation. This provides a simple way to express a potentially complex series using mathematical shorthand.

Arithmetic series

An arithmetic series is a sum of terms in a sequence where each term is increased by a constant difference. However, the sequence given in the problem \( \sum_{i=1}^{4} 2^{i} \) is not an arithmetic series; rather, it is closer to a geometric series because each term is multiplied by the same number.
An arithmetic series would look more like \(1 + 4 + 7 + 10\), where each term is increased by \(3\). In our exercise, each term is a power of \(2\), specifically, \(2^1, 2^2, 2^3, 2^4\). While it is not strictly arithmetic, understanding arithmetic series means grasping how terms build upon one another in a predictable pattern, much like the multiplication pattern shown but using addition.

Step by step solution

Working through problems step by step is a powerful way to understand complex mathematical concepts and processes. Let's break down the solution provided for the problem: 1. **Understand the Summation** The expression \(\sum_{i=1}^{4} 2^{i}\) notifies you to add up several values, specifically the powers of \(2\) from the first to the fourth. Each term is a distinct calculation that must be made individually.
2. **Calculate Each Term** You compute each power to find the individual terms: - \(2^1 = 2\) - \(2^2 = 4\) - \(2^3 = 8\) - \(2^4 = 16\) These numbers are fundamental components in deriving the final result.
3. **Add the Terms Together** Simply sum the terms: \(2 + 4 + 8 + 16\) Adding step by step ensures that results are accurate. Keeping track of each addition is key.
4. **Simplify the Sum** Execute the final addition to reach a simplified total of \(30\).Breaking problems into steps helps simplify the process and drives understanding. You can apply this method to various mathematical problems, ensuring clarity and reducing errors.