A sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = 2$ $a_{n+1} = 3a_n - c$ where $c$ is a constant - Edexcel - A-Level Maths Pure - Question 6 - 2011 - Paper 2

Given that $\sum_{i=1}^3 a_i = 0$, we need to find $a_3$ first.
We already have $a_1 = 2$ and we can use the expression for $a_2$:
[
a_2 = 6 - c.
]
Now, to find $a_3$:
[ a_3 = 3a_2 - c
= 3(6 - c) - c
= 18 - 3c - c
= 18 - 4c.
]
Now we can compute the sum:
[
a_1 + a_2 + a_3 = 2 + (6 - c) + (18 - 4c)
= 2 + 6 - c + 18 - 4c
= 26 - 5c.
]
Setting this equal to zero gives:
[ 26 - 5c = 0
\Rightarrow 5c = 26 \Rightarrow c = \frac{26}{5} = 5.2.
]