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

To find $x_3$, we use the expression we found for $x_2$:

$x_3 = a x_2 + 5$

Substituting $x_2 = a + 5$:

$x_3 = a(a + 5) + 5$

Expanding this gives:

$x_3 = a^2 + 5a + 5$

From earlier, we have:

$x_5 = a x_4 + 5$

Continuing this pattern, we find:

1. $x_4 = a x_3 + 5$; substituting $x_3 = a^2 + 5a + 5$ gives:

   $x_4 = a(a^2 + 5a + 5) + 5$

   Expanding will yield:

   $x_4 = a^3 + 5a^2 + 5a + 5$

2. Then substitute $x_4$ into $x_5$:

   $x_5 = a(a^3 + 5a^2 + 5a + 5) + 5$

Setting this equal to 41 leads to:

$a^4 + 5a^3 + 5a^2 + 5a + 5 = 41$

Subtracting 41 gives:

$a^4 + 5a^3 + 5a^2 + 5a - 36 = 0$

This is a polynomial equation in terms of $a$. Solving this equation using numerical methods or factoring will yield the possible values of $a$. The roots can be checked for suitable values.