Trees are planted in a triangle: 1 tree in the first row, 2 in the second, 3 in the third, and so on until all 3003 trees are used. How many rows are there?

Step-by-step solution

1. 1

   Recognise the arithmetic series. The row counts form the AP $1,2,3,\dots,n$ with $u_1=1$ and $d=1$, so the total number of trees is the triangular number $S_n=\frac{n(n+1)}{2}$

2. 2

   Set the sum equal to the stock of trees. $\frac{n(n+1)}{2}=3003\quad\Longrightarrow\quad n^2+n-6006=0$

3. 3

   Solve the quadratic. $n=\frac{-1+\sqrt{1+4\cdot 6006}}{2}=\frac{-1+\sqrt{24025}}{2}=\frac{-1+155}{2}=77$ The discriminant $24025=155^2$ is a perfect square, which is the sign that the problem was designed to close exactly.

4. 4

   Discard the negative root. The other root is $n=-78$, meaningless as a row count, so $n=77$.

5. 5

   Verify the total. $\frac{77\cdot 78}{2}=\frac{6006}{2}=3003$ trees, using every tree with none left over. The answer is A.