A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 4$ $a_{n+1} = k(a_n + 2)$ for $n > 1$ where $k$ is a constant - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 1

To find the two possible values of $k$, we first need to calculate $a_3$:

1. Using the expression found for $a_2$:
   $a_3 = k(a_2 + 2) = k(6k + 2) = 6k^2 + 2k$
2. Then, calculate the sum $\sum_{i=1}^{3} a_i$:
   $a_1 + a_2 + a_3 = 4 + 6k + (6k^2 + 2k) = 6k^2 + 8k + 4$
3. Set this sum equal to 2:
   $6k^2 + 8k + 4 = 2$
4. Rearranging gives:
   $6k^2 + 8k + 2 = 0$
5. Simplifying further:
   $3k^2 + 4k + 1 = 0$
6. Using the quadratic formula $k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a = 3$, $b = 4$, $c = 1$:
   $k = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} = \frac{-4 \pm \sqrt{16 - 12}}{6} = \frac{-4 \pm 2}{6}$
7. This leads to two possible values:
   $k = \frac{-2}{6} = -\frac{1}{3} \quad \text{and} \quad k = \frac{-6}{6} = -1.$
   Thus, the two possible values of $k$ are $-\frac{1}{3}$ and $-1$.