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Chapter 11: Problem 13

Evaluate. $$ \sum_{i=3}^{5} i(i+2) $$

Short Answer

Expert verified
The sum is 74.

Step by step solution

01

Understand the Problem

The problem asks to evaluate the sum of the expression \(i(i+2)\) from \(i = 3\) to \(i = 5\). This means we need to calculate the expression for each integer value of \(i\) between 3 and 5, and then find the resulting sum.
02

Evaluate for i = 3

Substitute \(i = 3\) into the expression \(i(i+2)\): \[3(3+2) = 3 \times 5 = 15\].
03

Evaluate for i = 4

Substitute \(i = 4\) into the expression \(i(i+2)\): \[4(4+2) = 4 \times 6 = 24\].
04

Evaluate for i = 5

Substitute \(i = 5\) into the expression \(i(i+2)\): \[5(5+2) = 5 \times 7 = 35\].
05

Calculate the Sum

Add the results from each step together to find the sum: \[15 + 24 + 35\]. Calculate this to get: \[15 + 24 = 39\]. Then, \[39 + 35 = 74\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series Evaluation
In mathematics, a series refers to the sum of the terms of a sequence. In our specific problem, we are tasked with evaluating a particular series noted with the summation symbol, \(\sum\). This is known as summation notation. It's a shorthand way to express the addition of a sequence of numbers. In the given exercise, the series is represented as \(\sum_{i=3}^{5} i(i+2)\). This requires us to find the value of the expression \(i(i+2)\) for each integer \(i\) from 3 to 5, then sum up these values. Here's how the process generally works:
  • Identify the expression to be evaluated, here it's \(i(i+2)\).
  • Determine the range of the variable, from 3 to 5 in this example.
  • Substitute each integer value within the range into the expression, compute it, and add them all together.
This method of evaluating a series is widely used to solve problems involving sequences in both pure and applied mathematics.
Algebraic Expressions
An algebraic expression combines numbers, variables, and arithmetic operations to represent a value. In our exercise, we have the expression \(i(i+2)\). Here, \(i\) is a variable that changes within a specified range (from 3 to 5). The structure of this expression indicates several operations:
  • The variable \(i\) itself, representing our current point in the range.
  • Addition inside the parentheses, \(i + 2\), which modifies the original variable.
  • Multiplication of the modified variable \(i + 2\) by the original variable \(i\).
Algebraic expressions are foundational to algebra, serving as a powerful tool for modeling real-world scenarios and solving equations. They allow us to represent complex relationships through straightforward mathematical symbols and operations. Understanding how to manipulate and evaluate these expressions is key to solving problems effectively.
Mathematical Operations
In this exercise, we worked through several mathematical operations to arrive at the solution. The operations needed include:
  • Substitution: Replacing the variable \(i\) with specific integer values from the range 3 to 5.
  • Addition: Performed within the parentheses to calculate \(i + 2\) for each value of \(i\).
  • Multiplication: After calculating \(i + 2\), we multiply the result by \(i\) to complete the evaluation of the expression.
Once each term is evaluated, you perform the final operation of addition to find the total sum: \(15 + 24 + 35 = 74\).It is essential to follow the order of operations, also known as "PEMDAS" (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to ensure correct results. This clear and methodical approach to handling mathematical operations helps in accurately evaluating expressions and solving problems.

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Most popular questions from this chapter

The number of otters born each year in a new aquarium forms a sequence whose general term is \(a_{n}=(n-1)(n+3) .\) Find the number of otters born in the third year and find the total number of otters born in the first three years.

Solve. Find the sum of the first seven terms of the sequence \(3, \frac{3}{2}, \frac{3}{4}, \dots\)

Show that \(\left(\begin{array}{l}{n} \\ {n}\end{array}\right)=1\) for any whole number \(n\)

Find the sum of the terms of each infinite geometric sequence. See Example 6 $$ 6,-4, \frac{8}{3}, \dots $$

a. Write the sum \(\sum_{i=1}^{7}\left(i+i^{2}\right)\) without summation notation. b. Write the sum \(\sum_{i=1}^{7} i+\sum_{i=1}^{7} i^{2}\) without summation notation. c. Compare the results of parts (a) and (b). d. Do you think the following is true or false? Explain your answer. $$\sum_{i=1}^{n}\left(a_{n}+b_{n}\right)=\sum_{i=1}^{n} a_{n}+\sum_{i=1}^{n} b_{n}$$
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