### Instruction: A particular geometric sequence has strictly decreasing terms. After the first term, each successive term is calculated by multiplying the previous term by $\frac{m}{7}$. If the first term of the sequence is positive, how many possible integer values are there for $m$?### Response: I need to find out how many values of $m$ make the sequence strictly decreasing.
That means I want the ratio $\frac{m}{7}$ to be less than 1, but also positive.
So I can write the inequality $0 < \frac{m}{7} < 1$.
To solve for $m$, I can multiply everything by 7, and get $0 < m < 7$.
This means that $m$ can be any positive integer between 0 and 7, but not including 0 or 7.
There are six such integers: 1, 2, 3, 4, 5, and 6.
So the answer is 6.