Observe the following expressions. [1(1/2)]^2 - [1/2]^2 = 2

Question

Observe the following expressions.

\(\begin{bmatrix} 1\frac{1}{2} \\[0.3em] \end{bmatrix}^2-\begin{bmatrix} \frac{1}{2} \\[0.3em] \end{bmatrix}^2\) = 2

\(\begin{bmatrix} 2\frac{1}{2} \\[0.3em] \end{bmatrix}^2-\begin{bmatrix} 1\frac{1}{2} \\[0.3em] \end{bmatrix}^2\) = 4

\(\begin{bmatrix} 3\frac{1}{2} \\[0.3em] \end{bmatrix}^2-\begin{bmatrix} 2\frac{1}{2} \\[0.3em] \end{bmatrix}^2\) = 6

Explain the common method in these using algebra.

Answer 1

n² – (n – 1)²

= n² – (n² – 2n + 1)

= n² – n² + 2n – 1

= 2n – 1

n² – (n – 1)² = 2n – 1