volume_mute
Algebra as a Language for Thinking
publish date: 2026/05/23 05:12:47.897936 UTC
volume_mute
Consider the arithmetic pattern below:
\(3^2\) is 1 bigger than \(2 \times 4\)
\(4^2\) is 1 bigger than \(3 \times 5\)
\(5^2\) is 1 bigger than \(4 \times 6\)
Which algebraic identity correctly generalises this pattern for any natural number \(n\)?
Correct Answer
\(n^2 = (n-1)(n+1) + 1\)
Explanation
The pattern generalises to \(n^2 = 1 + (n-1)(n+1)\). Expanding the right side: \((n-1)(n+1) = n^2 - 1\), so \(1 + (n^2 - 1) = n^2\). This demonstrates how algebra precisely captures arithmetic observations.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012