A sequence $x_1, x_2, x_3, \\ldots$ is defined by $x_1 = 1$ $x_{n} = ax_{n-1} + 5,$ $n > 1$ where $a$ is a constant - Edexcel - A-Level Maths Pure - Question 5 - 2012 - Paper 1

Continuing with the sequence, we find $x_3$ using:

$x_3 = ax_2 + 5.$

Substituting our expression for $x_2$: $x_3 = a(a + 5) + 5 = a^2 + 5a + 5.$

We need to express $x_5$ in terms of $a$. Using the previously found formulas for $x_2$ and $x_3$, we continue finding:

1. Calculate $x_4$: $x_4 = ax_3 + 5 = a(a^2 + 5a + 5) + 5 = a^3 + 5a^2 + 5a + 5.$

2. Calculate $x_5$: $x_5 = ax_4 + 5 = a(a^3 + 5a^2 + 5a + 5) + 5 = a^4 + 5a^3 + 5a^2 + 5a + 5.$

Setting $x_5 = 41$: $a^4 + 5a^3 + 5a^2 + 5a + 5 = 41.$

This simplifies to: $a^4 + 5a^3 + 5a^2 + 5a - 36 = 0.$

Now, we can use various methods (factorization, substitution, numerical methods) to find the possible values of $a$. Solving this polynomial may yield multiple roots which are candidates for $a$.