The 1st term of an arithmetic progression is $a$ and the common difference is $d$, where $d \ne 0$. The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms of a geometric progression.

1. Write down the expression for the 5th term

$u_5 = a + (5 - 1)d = a + 4d$

2. Write down the expression for the 15th term

$u_{15} = a + (15 - 1)d = a + 14d$

1. Set up an equation using the common ratio property of a geometric progression.

$\frac{a + 4d}{a} = \frac{a + 14d}{a + 4d}$

2. Cross-multiply and expand the equation.

$(a + 4d)^2 = a(a + 14d)$
$a^2 + 8ad + 16d^2 = a^2 + 14ad$

3. Simplify the equation to arrive at the given result, noting that $d \neq 0$.

$16d^2 = 6ad$
$16d^2 - 6ad = 0$
$2d(8d - 3a) = 0$
Since it is given that $d \ne 0$,
$8d - 3a = 0$
$3a = 8d$ (shown)

1. Express the common ratio in terms of $a$ and $d$.

The common ratio $r$ is the ratio of the second term to the first term of the geometric progression. Using the terms from part (i):
$r = \frac{a + 4d}{a}$

2. Substitute the relationship $3a = 8d$ to evaluate the common ratio.

From part (ii), we know that $3a = 8d$, which gives $d = \frac{3}{8}a$.
Substitute this into the expression for $r$:
$r = \frac{a + 4\left(\frac{3}{8}a\right)}{a}$
$r = \frac{a + 1.5a}{a}$
$r = \frac{2.5a}{a}$
Since $d \ne 0$, $a \ne 0$, so we can divide by $a$:
$r = 2.5$