Track / Stage 6 / 250 of 400 #1250 of 2000

Problem 1250

National olympiad, first round

Algebra Difficulty 6.4 Find the answer

A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $100$ cans, how many rows does it contain?

$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 11$

Your answer

Official solution

1. Let's denote the number of rows by $n$ . The number of cans in the $k$ -th row is given by $2k - 1$ . Therefore, the total number of cans in the display is the sum of the first $n$ odd positive integers.
2. The sum of the first $n$ odd positive integers is given by the formula:
$1 + 3 + 5 + \cdots + (2n - 1) = n^2$
This is a well-known result that can be derived by induction or by recognizing that the sum of the first $n$ odd numbers forms a perfect square.
3. Given that the total number of cans is 100, we set up the equation:
$n^2 = 100$
4. Solving for $n$ , we take the square root of both sides:
$n = \sqrt{100} = 10$
5. Therefore, the number of rows in the display is $n = 10$ .

The final answer is $\boxed{10}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.