Answer: \(t=-1.5\), and the common ratio is \(3\).

For an arithmetic progression with first term \(t\) and common difference \(1.5\), the \(n\)th term is

\(t+(n-1)(1.5)\).

Therefore

\(4\text{th term}=t+3(1.5)=t+4.5\),

\(8\text{th term}=t+7(1.5)=t+10.5\),

and

\(20\text{th term}=t+19(1.5)=t+28.5\).

These three terms are consecutive terms of a geometric progression. For three consecutive terms of a geometric progression, the square of the middle term is equal to the product of the other two terms. Hence

\((t+10.5)^2=(t+4.5)(t+28.5)\).

Expanding both sides,

\(t^2+21t+110.25=t^2+33t+128.25\).

Subtracting \(t^2+21t+110.25\) from both sides gives

\(0=12t+18\).

Thus

\(t=-\frac{18}{12}=-1.5\).

The three terms are then

\(t+4.5=3\), \(t+10.5=9\), \(t+28.5=27\).

So the common ratio of the geometric progression is

\(\dfrac{9}{3}=3\).