Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(\sum_{i=1}^{10} a_{i}=40\) and \(\sum_{i=1}^{10} b_{i}=50 .\) Calculate each of the following. $$ \sum_{p=0}^{9}\left(a_{p+1}-b_{p+1}\right) $$

Step-by-Step Solution

Answer

The result is -10.

Step 1: Understanding the Problem

We are asked to find the sum \( \sum_{p=0}^{9} (a_{p+1} - b_{p+1}) \). This requires us to calculate the difference between elements of two sequences, \( a \) and \( b \), from index 1 to 10 and then sum up these differences.

Step 2: Rewriting the Sum

Recognize that the index \( p \) starts at 0 and ends at 9, which means for each \( p \), we are taking \( a_{p+1} \) and \( b_{p+1} \). Thus, \( \sum_{p=0}^{9} (a_{p+1} - b_{p+1}) = (a_1 - b_1) + (a_2 - b_2) + \cdots + (a_{10} - b_{10}) \).

Step 3: Applying Sum Property

Use the property of sums that \( \sum (x_i - y_i) = \sum x_i - \sum y_i \). Therefore, \( \sum_{p=0}^{9} (a_{p+1} - b_{p+1}) = \sum_{i=1}^{10} a_i - \sum_{i=1}^{10} b_i \).

Step 4: Substituting Known Values

Substitute the known sums: \( \sum_{i=1}^{10} a_i = 40 \) and \( \sum_{i=1}^{10} b_i = 50 \). Thus, \( 40 - 50 \).

Step 5: Calculating the Final Result

Calculate the result by subtracting: \( 40 - 50 = -10 \).

Key Concepts

Summation Notation

Summation notation is a concise way of expressing the sum of a sequence of numbers. It uses the sigma symbol \( \Sigma \) to indicate that a set of numbers should be added together. In our exercise, we see \( \sum_{i=1}^{10} a_{i} \) and \( \sum_{i=1}^{10} b_{i} \), where the index \( i \) starts at 1 and ends at 10. Each \( a_{i} \) or \( b_{i} \) represents a term in the sequences \( a \) and \( b \), respectively. This notation is extremely useful for simplifying the representation of long sums, especially when dealing with sequences.

• The expression \( \sum_{i=1}^{10} a_{i} = a_1 + a_2 + ... + a_{10} \) involves 10 terms, each adding the terms from \( a_1 \) to \( a_{10} \).
• Similarly, \( \sum_{p=0}^{9} (a_{p+1} - b_{p+1}) \) implies calculating the difference of each term in sequences \( a \) and \( b \), and then summing these differences over the range provided.

Simplifying calculations using summation notation helps in performing complex arithmetic operations more efficiently. It also aids in recognizing patterns and properties, such as linearity, which is pivotal in solving our given task.

Series and Sequences

Series and sequences are foundational concepts in calculus and algebra. A sequence is an ordered list of numbers, and a series is the sum of the terms of a sequence. In this exercise, we deal with two sequences: \( a \) and \( b \).

• The sequence \( a \) is represented by terms \( a_1, a_2, ..., a_{10} \), and similarly, the sequence \( b \) is represented by terms \( b_1, b_2, ..., b_{10} \).
• When we sum the terms of these sequences, we form series. Series can describe infinite sums or finite sums, like in our exercise, where the series ends at a finite number.
• The sums of these series are given in the problem: \( \sum_{i=1}^{10} a_i = 40 \) and \( \sum_{i=1}^{10} b_i = 50 \).

A firm grasp of series and sequences involves understanding their definitions and how they are manipulated in mathematical expressions. This understanding is crucial for solving problems that require summing differences, as seen in our exercise.

Arithmetic Operations

Arithmetic operations form the building blocks of most mathematical equations and expressions. These operations include addition, subtraction, multiplication, and division. In the context of this exercise, the operation of subtraction is primarily used.

• When handling \( \sum_{p=0}^{9} (a_{p+1} - b_{p+1}) \), we apply the basic idea of subtracting terms of the sequence \( b \) from the sequence \( a \) one by one, before summing them up.
• Another key arithmetic operation used in the exercise is the addition of these differences, which is done smoothly thanks to the property that allows us to combine sums: \( \sum (x_i - y_i) = \sum x_i - \sum y_i \).
• This property simplifies complex expressions and is based on the linearity of addition and subtraction.

Understanding how arithmetic operations work in combination with properties of sums aids in solving mathematics problems efficiently. They allow us to break problems into smaller, manageable pieces, and ensure accuracy in calculations. In our exercise, this principle leads to finding that the overall sum of sequence differences is \(-10\).