A sequence $a_1, a_2, a_3,...$ is defined by a_1 = 1 a_{n+1} = \frac{k(a_n + 1)}{a_n}, \, n > 1 where $k$ is a positive constant - Edexcel - A-Level Maths Pure - Question 5 - 2017 - Paper 1

Given that $\sum_{r=1}^3 a_r = 10$, we calculate:

$a_1 + a_2 + a_3 = 1 + 2k + \frac{2k + 1}{2} = 10.$

Combining like terms, we have:

$1 + 2k + \frac{2k + 1}{2} = \frac{2}{2} + 2k + \frac{2k + 1}{2} = \frac{2 + 4k + 2k + 1}{2} = \frac{1 + 6k}{2} = 10.$

Multiplying both sides by 2 gives:

$1 + 6k = 20\Rightarrow 6k = 19\Rightarrow k = \frac{19}{6}.$

Thus, the exact value for $k$ is $\frac{19}{6}$.