How $L_n = L_{n-1} + n $ become $L_{n-2} + (n-1) +n $ and So on?

Question

I am currently reading 'concrete mathematics' of knuth. I don't know how

$L_n = L_{n-1} + n $

become

$L_{n-2} + (n-1) +n $

and finally

$L_0+1+2...+(n-2)+(n-1)+n $

can you please tell me?

Answer 1

From $L_n=L_{n-1}+n$ we can get $L_{n-1}=L_{n-2}+(n-1)$. Now substitute for $L_{n-1}$ in the previous equation.