AMC10 2004 B

AMC10 2004 B · Q10

AMC10 2004 B · Q10. It mainly tests Quadratic equations, Arithmetic sequences basics.

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?

杂货商摆放罐头，最顶层一行一个罐头，每下一行比上一行多两个罐头。如果展示中共有 100 个罐头，有多少行？

(A) 5 5

(B) 8 8

(D) 10 10

(E) 11 11

Answer

Correct choice: (D)

正确答案：（D）

Solution

(D) If there are $n$ rows in the display, the bottom row contains $2n-1$ cans. The total number of cans is therefore the sum of the arithmetic series $$ 1+3+5+\cdots+(2n-1), $$ which is $$ \frac{n}{2}\left[(2n-1)+1\right]=n^2. $$ Thus $n^2=100$, so $n=10$.

（D）如果陈列有 $n$ 行，那么最底下一行有 $2n-1$ 个罐子。因此罐子的总数是等差数列之和 $$ 1+3+5+\cdots+(2n-1), $$ 其和为 $$ \frac{n}{2}\left[(2n-1)+1\right]=n^2. $$ 因此 $n^2=100$，所以 $n=10$。

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