Problem 15 - 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. $$ \sum_{i=1}^{10}\left(a_{i}+b_{i}\right) $$

Step-by-Step Solution

Answer

The sum \( \sum_{i=1}^{10} (a_i + b_i) \) is 90.

Step 1: Understanding the Problem

We are given two separate summations, \( \sum_{i=1}^{10} a_{i} = 40 \) and \( \sum_{i=1}^{10} b_{i} = 50 \), and we need to find the value of \( \sum_{i=1}^{10}(a_{i} + b_{i}) \). This requires us to combine the individual sums to obtain the desired result.

Step 2: Rewriting the Target Expression

The expression \( \sum_{i=1}^{10}(a_{i} + b_{i}) \) can be rewritten by separating the sum of \(a_i\)s and \(b_i\)s. This can be expressed as: \( \sum_{i=1}^{10} a_i + \sum_{i=1}^{10} b_i \).

Step 3: Substituting Known Values

Using the given data, substitute \( \sum_{i=1}^{10} a_i = 40 \) and \( \sum_{i=1}^{10} b_i = 50 \) into the expression from the previous step. This gives us: \( 40 + 50 \).

Step 4: Calculating the Result

Now simply add the two numbers: \( 40 + 50 = 90 \). This is the value of \( \sum_{i=1}^{10} (a_i + b_i) \).

Key Concepts

Algebraic Manipulation

In mathematics, algebraic manipulation involves rearranging and simplifying expressions to make them easier to solve or understand. This can include combining like terms, factoring, or using properties of operations, such as addition or multiplication, to rearrange terms.
In our exercise, algebraic manipulation is used by recognizing that the addition of two separate sums can simplify the expression. We are asked to find the sum of two sequences, \( \sum_{i=1}^{10} (a_i + b_i) \).
To handle this, we first separate the expression into two distinct sums: \( \sum_{i=1}^{10} a_i + \sum_{i=1}^{10} b_i \).

• This step involves rearranging the terms within the summation to break it down into parts we already know the solution to.
• By rewriting the expression, we directly apply known information on each part of the problem and can therefore substitute the given values.

Applying algebraic manipulation effectively demands comprehension of the underlying arithmetic principles, helping in deriving simple solutions from seemingly complex problems.

Summation Properties

Summation is a fundamental concept in calculus that involves adding a sequence of numbers systematically. It has special properties that help simplify calculations and problem-solving. One key property used here is the distributive property of summation over addition.
This property allows us to split the sum of two sequences into separate sums:

• \( \sum_{i=1}^{10} (a_i + b_i) = \sum_{i=1}^{10} a_i + \sum_{i=1}^{10} b_i \)

This highlights how each term's individual sum can be calculated and combined, leveraging given information.
By recognizing and using this property, arithmetic tasks become manageable by breaking them into simpler, manageable parts.

Problem Solving in Calculus

Problem solving in calculus often requires a combination of understanding concepts, recognizing patterns, and applying known techniques. In this exercise, solving the problem relies primarily on comprehending the behavior of sums and their properties.
The steps include:

• Interpreting what the problem is asking, which forms the basis of a strategy to reach a solution.
• Simplifying or restructuring mathematical expressions using known rules, as demonstrated by rewriting the sum.

Finally, evaluative strategies are used, such as substitution and straightforward calculation of numerical results.
These steps illustrate how mathematical reasoning in calculus isn’t just about computation but about recognizing relationships and knowing how to use them effectively. Such skills are vital in tackling broader, more complex problems in mathematics and beyond.