Round #1094

Is the repeating decimal 0.999... equal to 1? State the exact value and justify using a geometric-series argument.

## Step 1: Understanding the Problem The problem asks us to determine if the repeating decimal 0.999... is equal to 1 and to justify this using a geometric-series argument. ## Step 2: Representing 0.999... as a Geometric Series 0.999... can be represented as an infinite geometric series: 0.9 + 0.09 + 0.009 + ..., where the first term \(a = 0.9\) and the common ratio \(r = 0.1\). ## Step 3: Applying the Formula for the Sum of an Infinite Geometric Series The sum \(S\) of an infinite geometric series is given by the formula \(S = \frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the common ratio, provided that \(|r| < 1\). ## Step 4: Calculating the Sum Substituting \(a = 0.9\) and \(r = 0.1\) into the formula gives us \(S = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1\). The final answer is: $\boxed{1}$

The repeating decimal 0.999... is equal to 1. To justify this using a geometric-series argument, consider the decimal 0.999... as an infinite series: 9/10 + 9/100 + 9/1000 + ... . This is a geometric series with the first term a = 9/10 and the common ratio r = 1/10. The sum of an infinite geometric series is given by the formula a / (1 - r), which is valid when |r| < 1. In this case, |r| = 1/10 < 1, so the series converges. Plugging in the values of a and r into the formula, we get: (9/10) / (1 - 1/10) = (9/10) / (9/10) = 1. Therefore, the repeating decimal 0.999... is equal to 1. Here is a simple C# code snippet to demonstrate this: ```csharp using System; public class GeometricSeries { public static double CalculateSeriesSum(double a, double r) { if (Math.Abs(r) >= 1) { throw new ArgumentException("The series diverges because |r| >= 1."); } return a / (1 - r); } public static void Main(string[] args) { double a = 9.0 / 10.0; // first term double r = 1.0 / 10.0; // common ratio double sum = CalculateSeriesSum(a, r); Console.WriteLine("The sum of the series is: " + sum); } } ``` This code calculates the sum of the geometric series using the formula a / (1 - r) and prints the result. When run, it will output "The sum of the series is: 1", confirming that 0.999... equals 1.