Problem 51 - Algebra for College Students

Question

Find the sum of the first 200 odd whole numbers. 40,000

Step-by-Step Solution

Answer

The sum of the first 200 odd whole numbers is 40,000.

Step 1: Understand the problem

We need to find the sum of the first 200 odd whole numbers. Odd numbers are numbers like 1, 3, 5, etc. These numbers follow a consistent pattern.

Step 2: Identify the formula

The sum of the first \( n \) odd numbers follows the formula: \( S = n^2 \) where \( n \) is the number of odd numbers to be added.

Step 3: Apply the formula

Since we are looking for the sum of the first 200 odd numbers, substitute \( n = 200 \) into the formula. Thus, \( S = 200^2 \).

Step 4: Calculate the sum

Calculate \( 200^2 \). The result is: \( 200 \times 200 = 40000 \). Thus, the sum of the first 200 odd numbers is 40,000.

Key Concepts

Odd Numbers

Odd numbers are integers that cannot be evenly divided by 2. They have a unique characteristic: when divided by 2, they leave a remainder of 1. Odd numbers play a key role in various mathematical problems and patterns. For example, the sequence of odd numbers starts as 1, 3, 5, 7, 9, and continues indefinitely.

These numbers can be easily identified by looking at their last digit. If a number ends in 1, 3, 5, 7, or 9, it is an odd number. This pattern remains consistent both in positive and negative numbers, such as -3, -5, -7, etc.

Understanding odd numbers is essential for solving problems related to sequences and series, like finding their sums and identifying patterns.

Series Formula

The concept of a series formula is vital when calculating the sum of odd numbers. For any sequence of numbers, a series formula provides a way to find the sum without manually adding each number.

When it comes to odd numbers, there's a simple and powerful formula at play: the sum of the first \( n \) odd numbers is given by \( S = n^2 \). This means if you want to sum up the first 200 odd numbers, you substitute \( n = 200 \) into the formula, getting \( S = 200^2 \), which equals 40,000.

This pattern makes it easy to calculate large sums quickly, saving time and energy compared to adding each number one by one. This formula has applications in various mathematical fields and helps illustrate how algebra can provide shortcuts to what might otherwise be long arithmetic processes.

Mathematical Patterns

In mathematics, recognizing patterns is a powerful skill that can simplify complex problems, such as finding the sum of numbers in a sequence. With odd numbers, this is evident in their consistent progression: each number is two more than the previous one.

This pattern helps to understand why the series formula \( S = n^2 \) works so effectively. By interpreting a group of n odd numbers in this way, you can see how they form a neat square area when visualized geometrically. For instance, arranging pebbles or blocks in a square illustrates how these numbers accumulate.

Such patterns are more than just mathematical curiosities; they are tools for solving real-world problems efficiently. Additionally, they promote logical thinking and reasoning skills, as they encourage recognizing regularities in different situations, ultimately making mathematics more intuitive and accessible.