The spelling bee

Thirty-two children entered the school spelling bee. In every round each child gets one word, and half of the children go through to the next round.

$32$ children; in each round every child gets $1$ word and half go through.

$n_0=32$ children; $n_k$ remain after round $k$: $n_{k+1}=\frac{n_k}{2}$.

So after round one there were sixteen, then eight, then four, then two. After five rounds only one child was left.

$32\to 16\to 8\to 4\to 2\to 1$: after $5$ rounds one is left, since $32=2^5$.

$n_5=\frac{32}{2^5}=1$.

I was one of the last two. The final word was "necessary". It has nine letters and I wrote ten, because I put in an extra c.

I was one of the last $2$. Final word "necessary": it has $9$ letters and I wrote $10$, that is $9<10$.

Final: $2$ left. "necessary": $9<10$.

The judges read out sixty-two words in all, and thirty-one children left the stage, everyone except the winner. I was one of the thirty-one, but I will never forget that word.

Words read out: $32+16+8+4+2=62$. Children out: $32-1=31$. I was one of the $31$, but I will never forget that word.

$32+16+8+4+2=62$, and $32-1=31$.