Connection Between Arithmetic Series and Trapezoids

Connection Between Arithmetic Series and Trapezoids

In this lesson, we explore a beautiful connection between algebra and geometry. Specifically, we examine how the formula for an arithmetic series is closely related to the formula for the area of a trapezoid.

1. Theory

An arithmetic series is the sum of the terms of an arithmetic sequence. Meanwhile, a trapezoid is a geometric shape with two parallel sides. Although these concepts come from different branches of mathematics, they share a surprisingly similar mathematical structure.

2. Concept Explanation

The key idea lies in the structure of their formulas. Both the arithmetic series formula and the trapezoid area formula use the average of two values, multiplied by a quantity, and divided by two. This similarity reveals how algebraic sums can be interpreted geometrically.

3. Steps or Formula

The formula for the sum of the first n terms of an arithmetic series is:

$$ S_n = \frac{(a_1 + a_n)n}{2} $$

The formula for the area of a trapezoid is:

$$ L = \frac{(s_1 + s_2) \times h}{2} $$

Notice the similarity:

• \(a_1\) and \(a_n\) correspond to the two parallel sides
• \(n\) corresponds to the height
• Both formulas divide the result by 2

4. Example Problem

Find the sum:

$$ 1 + 2 + 3 + \ldots + 99 $$

Step 1: Identify values.

$$ a_1 = 1, \quad a_{99} = 99, \quad n = 99 $$

Step 2: Substitute into the formula.

$$ S_{99} = \frac{(1 + 99) \times 99}{2} $$

Step 3: Simplify.

$$ S_{99} = \frac{100 \times 99}{2} = 4950 $$

5. Final Answer

The sum of the arithmetic series from 1 to 99 is 4950.

This sum can also be visualized geometrically as a set of stepped rectangles that form a trapezoid when rearranged. This elegant connection shows how algebra and geometry work together to reveal deeper mathematical patterns.