Problem 18 - Algebra for College Students

Question

A raffle is organized so that the amount paid for each ticket is determined by the number on the ticket. The tickets are numbered with the consecutive odd whole numbers \(1,3,5,7, \ldots\). Each contestant pays as many cents as the number on the ticket drawn. How much money will the raffle take in if 1000 tickets are sold? \(\$ 10,000\)

Step-by-Step Solution

Answer

The raffle will take in \$10,000 if 1000 tickets are sold.

Step 1: Understand the Ticket Numbering

The tickets are numbered using consecutive odd numbers starting from 1. These numbers are: 1, 3, 5, 7, and so on.

Step 2: Identify the N-th Odd Number

The n-th odd number can be found using the formula: \(2n-1\). Thus, the 1000th odd number would be \(2 \times 1000 - 1 = 1999\).

Step 3: Calculate the Sum of Odd Numbers

To find the total amount collected, sum all odd numbers from 1 to 1999. The sum of the first n odd numbers is \(n^2\). For 1000 tickets, it is \(1000^2 = 1000000\) cents.

Step 4: Convert Cents to Dollars

Since 1 dollar is 100 cents, the total amount in dollars is \(1000000 \div 100 = 10000\).

Key Concepts

Odd Numbers

Odd numbers are integers that are not divisible by 2. They have a remainder of 1 when divided by 2. Examples include 1, 3, 5, 7, etc. These numbers alternate with even numbers on the number line. Properties of Odd Numbers:

• When you add two odd numbers, the result is even. For example, 3 + 5 = 8.
• Multiplying two odd numbers always results in another odd number. For instance, 3 × 5 = 15.
• Subtracting one odd number from another results in an even number.

Odd numbers are fundamental in sequences and series exercises, such as finding sums or identifying patterns in math problems.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms, called the 'common difference,' is constant. In our raffle problem, the tickets are numbered with odd numbers, which form an arithmetic sequence.Features of an Arithmetic Sequence:

• The formula for the n-th term of an arithmetic sequence is: \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference.
• For consecutive odd numbers, the common difference \(d\) is 2, since each number differs from the previous by 2.

Arithmetic sequences are used to creative efficient ways of solving problems that involve summing consecutive numbers, such as our ticket prices.

Summation of Series

Summation of series involves adding up the terms in a sequence. In the raffle example, we need to sum the prices of all tickets, given by consecutive odd numbers from 1 to 1999. Understanding Series Summation:

• The sum of the first \(n\) odd numbers is \(n^2\). For example, if \(n = 7\), the sum is \(1 + 3 + 5 + 7 + 9 + 11 + 13 = 49\), or \(7^2\).
• This property can be used for quick calculations. It saves time compared to adding each number individually in contexts like finding the total raffle income.
• Summation techniques are a powerful tool for evaluating large series efficiently, especially when the series involves simple patterns.

Raffle Problem Solving

Raffle problem solving often involves finding creative ways to interpret and calculate scenarios based on given rules. In this exercise, the ticket pricing is determined by a numerical sequence.Steps to Solve Raffle Problems:

• Identify the series or pattern involved. This ensures you correctly interpret each ticket's cost.
• Use arithmetic sequence formulas to determine the number range and sum them if necessary. For odd numbered tickets, apply the fact that their sum is \(n^2\) for \(n\) tickets.
• Always convert results as needed, such as turning cents into dollars to get a practical financial total.

These problem-solving methods are not just limited to raffles; they are applicable to various real-world scenarios where logic and mathematics intersect.