Problem 48 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The sum is 34.

Step 1: Understanding the Summation

The expression \( \sum_{i=1}^{3} i 2^i \) represents the sum of a sequence where each term is given by \( i \cdot 2^i \). The variable \( i \) will take on integer values from 1 to 3.

Step 2: Calculating Each Term

Evaluate each term as \( i \) increases from 1 to 3:- For \( i = 1 \): \( 1 \cdot 2^1 = 2 \)- For \( i = 2 \): \( 2 \cdot 2^2 = 8 \)- For \( i = 3 \): \( 3 \cdot 2^3 = 24 \)

Step 3: Adding the Terms Together

Add the terms that have been calculated:\[ 2 + 8 + 24 = 34 \]

Step 4: Verifying Our Results

Review the calculations:- For \( i = 1 \): Confirmed \( 1 \cdot 2^1 = 2 \)- For \( i = 2 \): Confirmed \( 2 \cdot 2^2 = 8 \)- For \( i = 3 \): Confirmed \( 3 \cdot 2^3 = 24 \)Finally, the summed result \( 2 + 8 + 24 = 34 \) is correct.

Key Concepts

Understanding Sequences

A sequence is a list of numbers that are arranged in a specific order. In mathematics, sequences are usually defined by a rule or formula that determines which number comes next. Each number in the sequence is called a term. Sequences can go on forever or they can have a fixed number of terms. For example, the sequence of natural numbers 1, 2, 3, 4, 5, and so on, follows a simple incremental rule.
In the provided exercise, the sequence is determined by the expression \( i \cdot 2^i \), where \( i \) is an integer ranging from 1 to 3. It means for each integer \( i \), you multiply it by 2 raised to the power of \( i \). As a result, we have the sequence: 2, 8, and 24. Each of these numbers are determined using the formula given.

• Term 1: when \( i = 1 \), the term is \( 1 \cdot 2^1 = 2 \)
• Term 2: when \( i = 2 \), the term is \( 2 \cdot 2^2 = 8 \)
• Term 3: when \( i = 3 \), the term is \( 3 \cdot 2^3 = 24 \)

Understanding sequences lays the foundation for other mathematical concepts such as series and integer sequences.

Exploring Series

A series is what you get when you add up the terms of a sequence. Unlike a sequence, which lists terms, a series sums them up. If you imagine each term as a building block, a series is like stacking those blocks together.
In our example, the series is the sum of the terms we calculated from the sequence \( i \cdot 2^i \). The process of turning a sequence into a series involves using the summation symbol \( \Sigma \). The summation \( \sum_{i=1}^{3} i 2^i \) means you add up the terms from \( i = 1 \) to \( i = 3 \):

• First term: \( 2 \)
• Second term: \( 8 \)
• Third term: \( 24 \)

This gives us a series: \( 2 + 8 + 24 \), which sums to 34. Series play a crucial role in mathematics, especially in calculus and analysis, where they help to find patterns and sums over infinite lists of numbers.

Introduction to Integer Sequences

An integer sequence is a sequence where every term is an integer. Integers are whole numbers and can be positive, negative, or zero. Integer sequences are all around us, such as the sequence of odd numbers 1, 3, 5, 7, or the sequence of square numbers 1, 4, 9, 16.
In the exercise, the sequence \( i \cdot 2^i \) results in the terms 2, 8, and 24, which are all integers, thus forming an integer sequence. Understanding integer sequences is fundamental in knowing how numbers relate to each other and how they can be manipulated.
The power of integer sequences extends beyond pure mathematics. They appear in computer science, cryptography, and even music theory. Recognizing and working with integer sequences helps develop strong analytical skills, important for solving complex problems in various fields.