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Chapter 14: Problem 14

Evaluate. See Example 1 $$ \sum_{i=2}^{4} i(i-3) $$

Short Answer

Expert verified
The summation evaluates to 2.

Step by step solution

01

Understand the Summation

The problem involves evaluating the summation \( \sum_{i=2}^{4} i(i-3) \). This means we need to substitute each integer from 2 to 4 into the expression \( i(i-3) \) and add the results.
02

Substitute and Evaluate for Each Term

First, substitute \(i = 2\) into the expression: \[ 2 (2-3) = 2 \times (-1) = -2 \]Next, substitute \(i = 3\): \[ 3 (3-3) = 3 \times 0 = 0 \]Finally, substitute \(i = 4\): \[ 4 (4-3) = 4 \times 1 = 4 \]
03

Sum All Terms

Add the results of each evaluated term: \[ -2 + 0 + 4 = 2 \]
04

Write the Final Answer

The summation yields a final result of 2.

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Key Concepts

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

Evaluating Summations
In algebra, evaluating summations involves finding the sum of a sequence of numbers governed by a specific rule or expression. This process can often be seen as a simplification of a series of repeated arithmetic operations. For example, the summation notation \( \sum_{i=2}^{4} i(i-3) \) prompts us to apply the expression \( i(i-3) \) to each integer from 2 to 4, and then sum those results.

Follow these steps:
  • Identify the range of values for which the expression will be calculated. In this exercise, the values are \( i = 2, 3, 4 \).
  • Substitute each integer into the expression to calculate individual terms. For instance, substituting \(i=2\) yields \(2 \times (2-3) = -2\).
  • Sum all evaluated terms together. Using our example, adding \(-2, 0,\) and \(4\) results in \(2\).
If you remember these steps, evaluating summations quickly becomes a straightforward arithmetic activity.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations that represent a particular quantity. In the context of summations, an algebraic expression often defines the rule by which individual terms in the summation are calculated.

In our exercise, \(i(i-3)\) is the algebraic expression to be evaluated for each integer value in the range specified by the summation. Breaking it down:
  • \(i\) is a variable that changes according to the specified range鈥攈ere, 2 through 4.
  • \((i-3)\) involves a simple operation that shifts each \(i\) value by subtracting 3, giving a new integer to multiply by \(i\).
  • Multiplication of \(i\) and \((i-3)\) generates terms like \(-2, 0, \) and \(4\).
Understanding how to manipulate and evaluate algebraic expressions allows you to handle other calculations beyond just summations, enhancing your algebraic toolkit.
Summation Notation
Summation notation, often expressed with the Greek letter Sigma (\(\sum\)), provides a powerful shorthand for representing the sum of a sequence governed by an algebraic expression. It avoids lengthy addition of terms by using concise expressions.

Consider the segment \(\sum_{i=2}^{4} i(i-3)\):
  • The symbol \(\sum\) signifies that a sum will be calculated.
  • The subscript \(i=2\) indicates the starting value for \(i\).
  • The number 4 at the upper limit shows the ending value for \(i\).
  • The expression \(i(i-3)\) within the summation tells us what calculation to perform for each \(i\) from 2 to 4.
This notation is not only concise but also versatile, as it can be adapted to cover different ranges and expressions. Mastering summation notation empowers you to tackle more complex mathematical problems with ease.

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Most popular questions from this chapter

Given are the first three terms of a sequence that is either arithmetic or geometric If the sequence is arithmetic, find \(a_{1}\) and \(d\). If a sequence is geometric, find \(a_{1}\) and \(\bar{r}\) $$ 2,4,6 $$

Find the sum of the terms of each infinite geometric sequence. $$12,6,3, \dots$$

Solve. A ball is dropped from a height of 20 feet and repeatedly rebounds to a height that is \(\frac{4}{5}\) of its previous height. Find the total distance the ball covers before it comes to rest. (IMAGE CANNOT COPY)

Solve. Jose takes a job that offers a monthly starting salary of \(\$ 4000\) and guarantees him a monthly raise of \(\$ 125\) during his first year of training. Find the general term of this arithmetic sequence and his monthly salary at the end of his training.

Solve. Find the sum of the first ten terms of the sequence \(-4,1,6, \ldots, 41\) where 41 is the tenth term.
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