Variable as Generalization Math Example 2

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

Example 2

medium
Explain why (n+1)2−n2=2n+1(n+1)^2 - n^2 = 2n + 1 is true for every integer nn.

Solution

  1. 1
    Expand the left side: (n+1)2=n2+2n+1(n+1)^2 = n^2 + 2n + 1.
  2. 2
    Subtract n2n^2: n2+2n+1−n2=2n+1n^2 + 2n + 1 - n^2 = 2n + 1.
  3. 3
    The left side simplifies to 2n+12n + 1, which equals the right side for all nn.

Answer

Identity verified: (n+1)2−n2=2n+1(n+1)^2 - n^2 = 2n + 1 for all nn.
The variable nn represents any integer—it is a generalization, not a specific unknown. The identity shows that the difference between consecutive perfect squares is always an odd number.

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 3 easy

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

Example 4 medium

Write a general formula for the sum of the first [formula] positive integers.

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