| Dictionary definition | Usage | |
|---|---|---|
| A. To derive by reasoning or implication | E | We see smoke and infer fire. |
| B. To surmise | F | Given some utterance, a listener may infer from it all sorts of things which neither the utterance nor the utterer implied. |
| C. To point out | G | I waited all day to meet him. From this you can infer my zeal to see him. |
| D. To hint | H | She did not take part in the debate except to ask a question inferring that she was not interested in the debate. |


For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: