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Problem 3 Easy

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 number in the row is the sum of the two numbers directly above it. The first four rows are shown below.

What is the sum of the digits of the sum of the numbers in the 11th row?

  • A.

    11

  • B.

    13

  • C.

    14

  • D.

    16

  • E.

    17

Answer:D
Solution

Let S_k be the sum of k-th row.

Then S_k = 2 \cdot S_{k-1} - (10+1) + (10+1) = 2S_{k-1} for k \geq 3.

Since S_2 = 11, we have S_k = 11 \cdot 2^{k-2} for k \geq 2.

S_{11} = 11 \times 2^9 = 11 \times 512 = 5632

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