The Fascinating Mathematical Expression: 1/1×2 + 1/2×3 + 1/3×4 + … + 1/2019×2020
Introduction
In the realm of mathematics, there exist certain expressions that never cease to amaze and intrigue us. One such expression is the subject of our discussion today: 1/1×2 + 1/2×3 + 1/3×4 + … + 1/2019×2020. At first glance, this expression may appear daunting, but as we delve deeper, we shall uncover the underlying beauty and pattern behind it.
Breaking Down the Expression
Let us begin by dissecting the expression into its constituent parts. We have a series of fractions, each with a numerator of 1, and denominators that increment by 1, starting from 1 and going up to 2019. The numerators of these fractions, on the other hand, follow a pattern of incrementing by 1, starting from 2 and going up to 2020.
Pattern Recognition
As we examine the expression more closely, a pattern begins to emerge. Notice that each fraction can be rewritten as:
1/n × (n + 1)
Where n ranges from 1 to 2019. This rewriting reveals a crucial insight: each fraction is equivalent to the difference between two consecutive terms in a harmonic series.
Simplification and Evaluation
With this newfound understanding, we can simplify the expression by combining the fractions. We get:
1 - 1/2 + 1/2 - 1/3 + 1/3 - … + 1/2019 - 1/2020
Cancelling out the terms, we are left with:
1 - 1/2020
Evaluating this expression, we get:
2020/2020 - 1/2020 = 2019/2020
Conclusion
In this article, we have explored the intriguing mathematical expression 1/1×2 + 1/2×3 + 1/3×4 + … + 1/2019×2020. Through pattern recognition and simplification, we have uncovered the surprising result that the expression evaluates to 2019/2020. This exercise serves as a testament to the beauty and elegance of mathematics, where complexity can often be reduced to simplicity with the right insights.
