1. **State the problem:** We are given the sum of the first $n$ terms of a sequence as $$S_n = n(n+1)(n+2)$$ and need to find the 10th term, $a_{10}$. 2. **Recall the formula for the $n$th term:** The $n$th term of a sequence can be found by subtracting the sum of the first $n-1$ terms from the sum of the first $n$ terms: $$a_n = S_n - S_{n-1}$$ 3. **Write expressions for $S_n$ and $S_{n-1}$:** $$S_n = n(n+1)(n+2)$$ $$S_{n-1} = (n-1)n(n+1)$$ 4. **Find the $n$th term:** $$a_n = n(n+1)(n+2) - (n-1)n(n+1)$$ 5. **Factor out common terms:** $$a_n = n(n+1)[(n+2) - (n-1)]$$ 6. **Simplify inside the bracket:** $$a_n = n(n+1)(n+2 - n + 1) = n(n+1)(3) = 3n(n+1)$$ 7. **Calculate the 10th term:** $$a_{10} = 3 \times 10 \times 11 = 330$$ **Final answer:** The 10th term is $330$.