Mathematics

This is a geometric sequence problem where you need to find when the terms drop below a threshold. First, identify the common ratio by dividing consecutive terms—you'll see the sequence shrinks by the same factor each time. Then set up an inequality using the general term formula $a_n = a_1 \cdot r^{n-1}$ and solve for which term $n$ first falls below 0.001. The key is recognizing that exponential decay with a ratio between 0 and 1 eventually makes terms arbitrarily small.

This is a geometric sequence problem where you need to find when the terms drop below a threshold. First, identify the common ratio by dividing consecutive terms—you'll see the sequence shrinks by the same factor each time. Then set up an inequality using the general term formula $a_n = a_1 \cdot r^{n-1}$ and solve for which term $n$ first falls below 0.001. The key is recognizing that exponential decay with a ratio between 0 and 1 eventually makes terms arbitrarily small.