Math Problem Statement

2021 × 2022 × 2023 × [1 4 × 2 5 × 3 6 × 4 7 × 5 8 × 6 9 × … × 2017 2020 × 2018 2021 × 2019 2022 × 2020 2023 ].

Solution

The expression you've provided is a large product involving two parts. Let's break it down step by step:

  1. The first part is:

2021×2022×20232021 \times 2022 \times 2023

  1. The second part is a long product, which appears to be a sequence of fractions. This pattern seems to pair numbers in a ratio, from 1/4 to 2020/2023. The second part can be written as:

14×25×36××20202023\frac{1}{4} \times \frac{2}{5} \times \frac{3}{6} \times \cdots \times \frac{2020}{2023}

Now, the key to simplifying this is to recognize that many terms will cancel out.

Simplifying the second part:

The sequence 14,25,36,,20202023\frac{1}{4}, \frac{2}{5}, \frac{3}{6}, \dots, \frac{2020}{2023} cancels out all the terms in the numerator except the very first and all terms in the denominator except the last few. Let's write this out:

The numerator will be: 1×2×3××2020=2020!1 \times 2 \times 3 \times \cdots \times 2020 = 2020!

The denominator will be: 4×5×6××2023=2023!4 \times 5 \times 6 \times \cdots \times 2023 = 2023!

Therefore, the second part simplifies to:

2020!2023!\frac{2020!}{2023!}

But, by the properties of factorials: 2020!2023!=12021×2022×2023\frac{2020!}{2023!} = \frac{1}{2021 \times 2022 \times 2023}

Final Simplification:

Multiplying the first and second parts together:

(2021×2022×2023)×12021×2022×2023=1(2021 \times 2022 \times 2023) \times \frac{1}{2021 \times 2022 \times 2023} = 1

Thus, the value of the entire expression is:

1\boxed{1}

Would you like more details or have any questions about this solution?


Here are 5 related questions:

  1. What are factorials and how are they calculated?
  2. Can we generalize this type of product for other similar patterns?
  3. How does cancellation of terms work in a large product of fractions?
  4. What are some common applications of factorials in mathematics?
  5. How does the concept of limits apply to factorials?

Tip: When dealing with large products or sequences, always check if terms can cancel each other to simplify the calculation.

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Math Problem Analysis

Mathematical Concepts

Factorials
Sequences and Series
Simplification of Fractions

Formulas

n! (Factorial)
Product of Fractions

Theorems

Factorial Simplification Theorem
Cancellation of Terms in Product Sequences

Suitable Grade Level

Grades 11-12 or Undergraduate