Variable as Generalization Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
Write a general formula for the sum of the first nn positive integers.

Solution

  1. 1
    The sum 1+2+3+⋯+n1 + 2 + 3 + \cdots + n is given by S=n(n+1)2S = \frac{n(n+1)}{2}.
  2. 2
    Here nn represents any positive integer—a generalization.

Answer

S=n(n+1)2S = \frac{n(n+1)}{2}
This formula works for any positive integer nn. The variable nn generalizes over all possible list lengths, not a single unknown.

About Variable as Generalization

A variable standing for any arbitrary member of a specified set, used to express statements that hold universally.

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More Variable as Generalization Examples

Example 1 easy

Show that [formula] holds for [formula] and for [formula].

Example 2 medium

Explain why [formula] is true for every integer [formula].

Example 3 easy

Is [formula] a generalization or an equation to solve?

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