Regroup terms.
\[\begin{aligned}&an=an(n+1)-1\\&a\times 1=1.\end{aligned}\]
Simplify \(a\times 1\) to \(a\).
\[\begin{aligned}&an=an(n+1)-1\\&a=1.\end{aligned}\]
Break down the problem into these 2 equations.
\[an=an(n+1)-1\]
\[an=a\]
Solve the 1st equation: \(an=an(n+1)-1\).
Expand.
\[an=a{n}^{2}+an-1\]
Cancel \(an\) on both sides.
\[0=a{n}^{2}-1\]
Add \(1\) to both sides.
\[1=a{n}^{2}\]
Divide both sides by \({n}^{2}\).
\[\frac{1}{{n}^{2}}=a\]
Switch sides.
\[a=\frac{1}{{n}^{2}}\]
\[a=\frac{1}{{n}^{2}}\]
Solve the 2nd equation: \(an=a\).
Subtract \(an\) from both sides.
\[0=a-an\]
Factor out the common term \(a\).
\[0=a(1-n)\]
Divide both sides by \(1-n\).
\[0=a\]
Switch sides.
\[a=0\]
\[a=0\]
Collect all solutions.
\[a=\frac{1}{{n}^{2}},0\]
a=1/n^2,0
