AMC10 2025 B

AMC10 2025 B · Q3

AMC10 2025 B · Q3. It mainly tests Sequences & recursion (algebra), Arithmetic misc.

A Pascal-like triangle has $10$ as the top row and $10$ followed by $1$ as the second row. In each subsequent row the first number is $10$, the last number is $1$, and, as in the standard Pascal Triangle, each other in the row is the sum of the two numbers directly above it. The first four rows are shown below. \[\large{10}\] \[\large{10}\qquad\large{1}\] \[\large{10}\qquad\large{11}\qquad\large{1}\] \[\large{10}\qquad\large{21}\qquad\large{12}\qquad\large{1}\] What is the sum of the digits of the sum of the numbers in the 11th row?

一个类似帕斯卡三角形的三角形，第一行是10，第二行是10后面跟着1。后续每行的第一个数是10，最后一个数是1，其余每个数是其正上方两个数的和，就像标准帕斯卡三角形一样。下面展示了前四行。 \[\large{10}\] \[\large{10}\qquad\large{1}\] \[\large{10}\qquad\large{11}\qquad\large{1}\] \[\large{10}\qquad\large{21}\qquad\large{12}\qquad\large{1}\] 第11行的数字之和的各位数字之和是多少？

(A) 11 11

(B) 13 13

(D) 16 16

(E) 17 17

Answer

Correct choice: (D)

正确答案：（D）

Solution

If we find the sum for each rows, I'll only do the first 5, we find that row 1 has sum $10$, row 2 has sum $11$, row 3 has sum $22$, row 4 has sum $44$, and row 5 has sum $88$. we can see that excluding the first row each row has sum $11*2^{n-2}$ where $n$ is the row. This means the 11th row will have sum $11*2^9=11*512=5632$. The sum of the digits of $5632$ is $16$, so the answer is $\boxed{\textbf{(D) }16}$. Notice that the sum of each row is the slightly different than the normal Pascal's triangle, so the sum of the numbers is similar. Here's a proof of why each number in the normal Pascal's triangle is twice the previous number. To prove that the sum of each row is $2$ times the previous row, use the equation ${(1+1)}^n$ with the normal Pascal's triangle and use the binomial theorem from there.

如果我们计算每行的和，我只计算前5行，发现第1行和为10，第2行和为11，第3行和为22，第4行和为44，第5行和为88。可以看出，除第一行外，每行和为11*2^{n-2}，其中n是行号。这意味着第11行和为11*2^9=11*512=5632。5632的各位数字之和是16，所以答案是\boxed{\textbf{(D) }16}。注意，每行和与标准帕斯卡三角形略有不同，但类似。这里证明为什么每行和是前一行的2倍：在标准帕斯卡三角形中使用二项式定理{(1+1)}^n。

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