Problem 18 - 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. $$ \sum_{q=1}^{10}\left(a_{q}-b_{q}-q\right) $$

Step-by-Step Solution

Answer

The result is -65.

Step 1: Understand the Expression

The expression we need to evaluate is \(\sum_{q=1}^{10}(a_q - b_q - q)\). This means we sum up the values of \(a_q - b_q - q\) as \(q\) varies from 1 to 10.

Step 2: Break Down the Sum

By the linearity of summation, we can split the summation into three separate sums: \(\sum_{q=1}^{10} a_q - \sum_{q=1}^{10} b_q - \sum_{q=1}^{10} q\).

Step 3: Use Given Information

We know \(\sum_{i=1}^{10} a_i = 40\) and \(\sum_{i=1}^{10} b_i = 50\), so we substitute these into our expression, obtaining \(40 - 50 - \sum_{q=1}^{10} q\).

Step 4: Calculate the Sum of Integers

The sum of integers from 1 to \(n\) is given by the formula \(\frac{n(n+1)}{2}\). For \(n=10\), this sum is \(\frac{10 \times 11}{2} = 55\).

Step 5: Plug Values into the Expression

Substitute all the known values back into the expression: \(40 - 50 - 55\).

Step 6: Simplify and Find the Result

First calculate \(40 - 50 = -10\), then subtract 55, resulting in \(-10 - 55 = -65\).

Key Concepts

Calculus

Calculus is a branch of mathematics that focuses on the study of change and motion through derivatives and integrals. One of its core ideas is analyzing functions and their rates of change.
When dealing with sequences and series, calculus helps us understand how these numbers behave in a broader sense, especially as they approach infinity or change over time. For finite sums, like those in our exercise, calculus principles underpin the logic of breaking down expressions into component parts and simplifying them.

• Derivatives help us find the rate at which quantities increase or decrease.
• Integrals are used to find areas under curves, allowing us to measure aggregate change.
• Calculus is fundamental in finding and utilizing summation formulas, giving us tools to explore complex mathematical series.

Thus, calculus plays a pivotal role even when it's not explicitly calculated, forming the backbone of the techniques and understanding needed to solve problems involving series and sequences.

Series and Sequences

In mathematics, a sequence is an ordered list of numbers following a particular pattern, while a series is the sum of the terms of a sequence. Understanding the difference between a series and a sequence is fundamental to solving problems like the one in our exercise.
Sequences can be finite or infinite, and they can follow simple patterns such as arithmetic or geometric progressions. A simple sequence might be \(1, 2, 3,\dots, n\), where each number is obtained by adding a fixed constant to the last one. When these terms are summed, we get a series.

• Arithmetic series: The difference between successive terms is constant. The sum of the first \(n\) terms is calculated using the formula \frac{n}{2}(a_1 + a_n)\.
• Geometric series: Each term is a fixed multiple of the previous one. The sum can be calculated if the ratio is less than one.
• Summation notation \(\sum\) is a concise way to express the addition of a sequence of numbers.

In our example, \(\sum_{q=1}^{10}\left(a_q - b_q - q\right)\), represents a series formed from the differences in sequences and their indices.

Linear Summation

Linear summation is the process of adding a sequence of numbers linearly, often simplified by using properties like the linearity of addition. This is a crucial technique used in the original exercise.
By invoking the principle of linearity, we can break complex expressions into more manageable parts. This allows us to handle each component independently, reducing errors and simplifying calculations. The decomposition in our example into separate sums of \(a_q\), \(b_q\), and \(q\) relies on this concept.

• Linearity of summation allows us to rearrange sums: \[\sum (a_i + b_i) = \sum a_i + \sum b_i\.\]
• This property holds both for finite sums and in many cases for infinite series as well.
• In practical terms, this means we can compute complex series by calculating simpler components.

Practicing linear summation helps in understanding complicated summation problems, providing a clearer pathway to the correct solution.