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Chapter 9: Problem 89

In Exercises 85-96, find the sum. \( \displaystyle \sum_{i=0}^{4} i^2 \)

Short Answer

Expert verified
The required sum is 30.

Step by step solution

01

Understand the sum notation

The given expression, \( \displaystyle \sum_{i=0}^{4} i^2 \), means that we need to find the sum of the squares of all integers from 0 to 4. This is expressed in the form of a definite series, where the variable 'i' is the index of summation.
02

Apply the sum formula

The sum of squares of the first n numbers is given by the formula \( \frac{n(n + 1)(2n + 1)}{6} \). However since here, we are summing from 0 instead of 1, we need to subtract the zeroth term (which is 0) from the sum upto 4.
03

Calculation

Substituting the value n = 4 into the formula, the sum is \( \frac{4 \cdot 5 \cdot 9}{6} \). Using simple arithmetic, this simplifies to 30.

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Key Concepts

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

Sum of Squares
The concept of the sum of squares refers to adding together the squares of a set of numbers. In mathematical terms, this is often expressed using summation notation as seen with \( \displaystyle \sum_{i=0}^{4} i^2 \). The expression requires you to square each number individually and add those squared values.

In our example, we have integers from 0 to 4. Here are their squares:
  • 0 squared is 0
  • 1 squared is 1
  • 2 squared is 4
  • 3 squared is 9
  • 4 squared is 16
After squaring, we sum these values: 0, 1, 4, 9, and 16. The sum of these squares is 30.

Understanding the sum of squares is useful, especially in statistics and data analysis, where variations from a mean value often are expressed this way.
Index of Summation
The index of summation is a critical component when dealing with sum notation, represented usually by a variable within the summation symbol. In our expression \( \sum_{i=0}^{4} i^2 \), the 'i' is the index of summation.

This index tells you which values to consider in the series and ensures each is counted once. In our example, 'i' runs from 0 to 4. Hence:
  • The first value is found by letting \( i = 0 \)
  • The second value by letting \( i = 1 \)
  • Continuing this way until \( i = 4 \)
The control provided by the index of summation allows us to systematically apply operations 鈥� in this case, squaring 鈥� to each integer and sum them efficiently. Through this index, the expression clearly delineates the bounds of calculation.
Definite Series
A definite series refers to the sum of terms defined by a particular expression, where the series has specific lower and upper limits. This is different from an indefinite series, which continues without a terminal point.

In \( \sum_{i=0}^{4} i^2 \), we see a definite series because the index 'i' takes on values in a fixed range, from 0 to 4 inclusive.
  • The lower limit is 0, meaning 'i' starts at 0
  • The upper limit is 4, meaning 'i' ends at 4
Such series provide the advantage of clear boundaries, making them perfect for calculations that require finiteness. Understanding definite series helps methodically resolve many mathematical problems where comprehending the bounds of calculation is essential.

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Most popular questions from this chapter

Three points that are not collinear determine three lines. How many lines are determinedby nine points, no three of which are collinear?

In Exercises 79 - 86, solve for \( n \). \( _nP_5 = 18 \cdot _{n - 2} P_4 \)

A U.S. Senate Committee has \( 14 \) members. Assuming party affiliation was not a factor in selection, how many different committees were possible from the \( 100 \) U.S. senators?

Consider a group of people. (a) Explain why the following pattern gives the probabilities that the people have distinct birthdays. \( n = 2: \dfrac{365}{365} \cdot \dfrac{364}{365} = \dfrac{365 \cdot 364}{365^2} \) \( n = 3: \dfrac{365}{365} \cdot \dfrac{364}{365} \cdot \dfrac{363}{365} = \dfrac{365 \cdot 364 \cdot 363}{365^3} \) (b) Use the pattern in part (a) to write an expression for the probability that \( n = 4 \) people have distinct birthdays. (c) Let \( P_n \) be the probability that the \( n \) people have distinct birthdays. Verify that this probability can be obtained recursively by \( P_1 = 1 \) and \( P_n = \dfrac{365 - (n - 1)}{365} P_{n - 1} \). (d) Explain why \( Q_n = 1 - P_n \) gives the probability that at least two people in a group of \( n \) people have the same birthday. (e) Use the results of parts (c) and (d) to complete the table. (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than \( \dfrac{1}{2} \)? Explain.

In Exercises 79 - 86, solve for \( n \). \( _{n + 1} P_3 = 4 \cdot _nP_2 \)
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