Let's denote the first term of the arithmetic progression as \(a\) and the common difference as \(d\).

The \(n\)th term of an arithmetic progression is given by the formula:

T_n = a + (n - 1)d.\)

Given that the 9th term is five times the 5th term, we can write the relationship as:

T_9 = 5T_5.\)

Substituting the formulas for \(T_9\) and \(T_5\), we get:

a + 8d = 5(a + 4d).\)

Now, simplify the equation:

a + 8d = 5a + 20d.\)

Subtracting \(a\) from both sides:

8d = 4a + 20d.\)

Subtracting \(20d\) from both sides:

-12d = 4a.\)

Dividing both sides by \(4\):

3a + 5d = 0.\)