Question 3

Explanation

Given the geometric sequence $\frac{3}{2}, p, 6, q$ with a positive ratio.

Let the first term be $U_1 = a = \frac{3}{2}$ and the ratio be $r > 0$.

From $U_3 = 6$, we get

Since $r > 0$, then $r = 2$.

Statement $\text{(1)}$: The ratio of the sequence is $2$.

True. From the calculation above, we get $r = 2$.

Calculate the values of $p$ and $q$

Statement $\text{(2)}$: The value of $pq$ is $72$.

False. From the calculation above, $p = 3$ and $q = 12$, so

So the value of $pq$ is $36$, not $72$.

Calculate the sixth term

Statement $\text{(3)}$: The difference between the sixth and second term is $45$.

True. From the calculation above, $U_6 = 48$ and $U_2 = 3$, so

Statement $\text{(4)}$: Every term in the sequence is an even number.

False. From the calculation above, $U_1 = \frac{3}{2}$ is not an integer, and $U_2 = 3$ is an odd number. So not all terms are even numbers.

Therefore, the true statements are $\text{(1)}$ and $\text{(3)}$.