 (1+ \dfrac{ 1  }{ 2020  }  + \dfrac{ 1  }{ 2021  }  )(1+ \dfrac{ 1  }{ 2020  }  + \dfrac{ 1  }{ 2021  }  + \dfrac{ 1  }{ 2022  }  )-(1+ \dfrac{ 1  }{ 2020  }  + \dfrac{ 1  }{ 2021  }  + \dfrac{ 1  }{ 2022  }  )( \dfrac{ 1  }{ 2020  }  + \dfrac{ 1  }{ 2021  }  ) 

We are tasked with simplifying the following expression:

\[
\left( 1 + \frac{1}{2020} + \frac{1}{2021} \right)\left( 1 + \frac{1}{2020} + \frac{1}{2021} + \frac{1}{2022} \right) - \left( 1 + \frac{1}{2020} + \frac{1}{2021} + \frac{1}{2022} \right) \left( \frac{1}{2020} + \frac{1}{2021} \right)
\]

### Step 1: Introduce shorthand
Let’s introduce variables to simplify the expression:
- Let \( a = 1 + \frac{1}{2020} + \frac{1}{2021} \)
- Let \( b = \frac{1}{2022} \)

This allows us to rewrite the expression as:
\[
(a)(a + b) - (a + b)\left( \frac{1}{2020} + \frac{1}{2021} \right)
\]

### Step 2: Factor the expression
Notice that \( (a + b) \) is common to both terms, so we can factor it out:

\[
(a + b) \left( a - \left( \frac{1}{2020} + \frac{1}{2021} \right) \right)
\]

### Step 3: Simplify the inner part
Now, simplify the term inside the parentheses:

\[
a - \left( \frac{1}{2020} + \frac{1}{2021} \right) = \left( 1 + \frac{1}{2020} + \frac{1}{2021} \right) - \left( \frac{1}{2020} + \frac{1}{2021} \right) = 1
\]

Thus, the expression reduces to:

\[
(a + b) \cdot 1 = a + b
\]

### Step 4: Substitute back the values of \( a \) and \( b \)
Recall that:
\[
a = 1 + \frac{1}{2020} + \frac{1}{2021}, \quad b = \frac{1}{2022}
\]
Substituting these back, we get:
\[
a + b = 1 + \frac{1}{2020} + \frac{1}{2021} + \frac{1}{2022}
\]

Thus, the simplified expression is:

\[
1 + \frac{1}{2020} + \frac{1}{2021} + \frac{1}{2022}
\]

Would you like further clarification or details?

Here are five related questions to expand on this:
1. How would the result change if the denominators were consecutive even numbers?
2. Can you expand this problem to include more terms like \( \frac{1}{2023} \)?
3. What happens if we remove the first "1" in each part of the expression?
4. Can you generalize this problem for any set of consecutive numbers?
5. How would this expression behave as the numbers in the denominators approach infinity?

**Tip:** When simplifying complex algebraic expressions, look for common factors that can be factored out to make the calculation easier.

(1+ \dfrac{1}{2020}  + \dfrac{1}{2021})(1+ \dfrac{1}{2020}  + \dfrac{1}{2021}  + \dfrac{1}{2022})-(1+ \dfrac{1}{2020}  + \dfrac{1}{2021}  + \dfrac{1}{2022})(\dfrac{1}{2020}  + \dfrac{1}{2021})

Simplify the algebraic expression involving fractions (1 + 1/2020 + 1/2021)(1 + 1/2020 + 1/2021 + 1/2022) - (1 + 1/2020 + 1/2021 + 1/2022)(1/2020 + 1/2021) with a clear step-by-step solution. This problem covers algebraic simplification and factoring techniques, suitable for students in Grades 10-12.

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