AMC 8 Daily Practice Round 11

Complete problem set with solutions and individual problem pages

Problem 17 Medium

As shown in the figure, the ancient Greeks often used small pebbles to form various shapes on the sand to study numbers. For example, the numbers 1, 5, 12, 22, \cdots are known as pentagonal numbers. What is the 7-th pentagonal number?

  • A.

    35

  • B.

    51

  • C.

    58

  • D.

    70

  • E.

    72

Answer:D

The sequence of pentagonal numbers is formed as follows:

\text{1st pentagonal number: } 1, \text{2nd pentagonal number: } 1 + 4 = 5, \text{3rd pentagonal number: } 1 + 4 + 7 = 12, \text{4th pentagonal number: } 1 + 4 + 7 + 10 = 22.

Continuing the pattern, we calculate:

\text{5th pentagonal number: } 1 + 4 + 7 + 10 + 13 = 35, \text{6th pentagonal number: } 1 + 4 + 7 + 10 + 13 + 16 = 51, \text{7th pentagonal number: } 1 + 4 + 7 + 10 + 13 + 16 + 19 = 70.

Thus, the 7th pentagonal number is \boxed{70}.

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