### Instruction: The terms $140, a, \frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$?### Response: I know that a geometric sequence is one where each term is obtained by multiplying the previous term by a constant ratio.
So, to find the value of $a$, I need to find the ratio and then use it to get from the first term to the second term.
I can find the ratio by dividing any term by the previous term. For example, the ratio is $\frac{a}{140}$ or $\frac{45/28}{a}$.
Since the ratio is constant, these two expressions must be equal. So I can set up an equation: $\frac{a}{140} = \frac{45/28}{a}$.
To solve for $a$, I can cross-multiply and get $a^2 = \frac{45}{28} \cdot 140$.
Taking the square root of both sides, I get $a = \pm \sqrt{\frac{45}{28} \cdot 140}$.
However, I know that $a$ is positive, so I only consider the positive root.