Closed form expression of a summation

Question

My prof started out with the following summation: \begin{equation} \sum_{i=0}^{k}i = \frac{k(k+1)}{2} \end{equation} Which is all fine and dandy, however the summation we want to find the closed form of is \begin{equation} \sum_{i=0}^{k}(3i+1) = ? \end{equation} Which I did the following to: \begin{equation} \sum_{i=0}^{k}(3i+1) = 3\frac{k(k+1)}{2} + 1 \end{equation}

However her answer was: \begin{equation} \sum_{i=0}^{k}(3i+1) = 3\frac{k(k+1)}{2} + (k+1) \end{equation}

Is there some summation rule I am missing? How did that $(k+1)$ get added towards the end?

Answer 1

I assume the second line should have brackets around the $(3i+1)$. Then:

$$\sum_{i=0}^k(3i+1)=\sum_{i=0}^k3i+\sum_{i=0}^k1$$

As summation is linear, we can apply your prof's formula to the first term.

Answer 2

$$\begin{equation} \sum_{i=0}^{k}1 = k+1 \end{equation}$$

Answer 3

You have an error in the sum : $\sum_0^k 1 =k+1$ and not $1$