Math Problem Statement

Tính giá trị của biểu thức: A = 1 × 1/2 × 1/3 × 1/4 ... × 1/2023

Solution

The expression in the image is asking to evaluate the value of the following product:

A=1×12×13×14××12023A = 1 \times \frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} \times \dots \times \frac{1}{2023}

This is a product of fractions, starting from 12\frac{1}{2} and continuing with 13\frac{1}{3}, 14\frac{1}{4}, and so on, up to 12023\frac{1}{2023}.

We can express this product as:

A=k=220231kA = \prod_{k=2}^{2023} \frac{1}{k}

This is equivalent to:

A=12×3×4××2023A = \frac{1}{2 \times 3 \times 4 \times \dots \times 2023}

This denominator is just the factorial of 2023, divided by 1 (since 1×2×3××2023=2023!1 \times 2 \times 3 \times \dots \times 2023 = 2023!).

Therefore:

A=12023!A = \frac{1}{2023!}

So the value of the expression is 12023!\frac{1}{2023!}, which is a very small number.

Let me know if you need further details or clarification!


Here are some follow-up questions:

  1. What is the general form for such factorial-related products?
  2. How can you simplify similar expressions with products and factorials?
  3. What are the applications of large factorials in combinatorics and probability?
  4. How do you approximate very large factorials?
  5. What are some interesting properties of factorials in mathematical analysis?

Tip: Factorials grow extremely fast as the number increases. For instance, 2023!2023! is much larger than 100!100!, which is already too large to compute directly without special software or methods.

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Math Problem Analysis

Mathematical Concepts

Product of Fractions
Factorials
Sequences

Formulas

A = 1 / (2 * 3 * 4 * ... * 2023)
A = 1 / 2023!

Theorems

Factorial Formula

Suitable Grade Level

Grades 9-12