Solving a recurrence relation (homework)

Question

Solve the following recurrence relation:

$$l(n+1)=l(n)+π(4+2cn), \quad n=0,1,2,3,\ldots$$

Answer 1

You can rewrite the equation as: $$ l(n + 1) - l(n) = \pi(4 + 2 c n) $$ Thus: $$ \begin{align*} \sum_{0 \le k < n} (l(k + 1) - l(k)) &= \sum_{0 \le k < n} \pi(4 + 2 c k) \\ l(n) &= l(0) + \sum_{0 \le k < n} \pi(4 + 2 c k) \end{align*} $$ Your recurrence is linear, al right.