AMC 8 Daily Practice - The Rule of Arithmetic Sequences

Complete problem set with solutions and individual problem pages

Problem 7 Easy

Lily wants to distribute 210 chess pieces into several boxes. The first box contains 1 piece, the second box contains 2 pieces, the third box contains 3 pieces, and so on. The pieces are exactly when placed in this manner. How many boxes does Lily use?

  • A.

    18

  • B.

    19

  • C.

    20

  • D.

    21

  • E.

    22

Answer:C

Assume Lily uses n boxes.

The total number of pieces follows the arithmetic series:   1 + 2 + 3 + \dots + n = 210

Using the arithmetic series sum formula:   \frac{n(n+1)}{2} = 210

Multiply both sides by 2:   n(n+1) = 420

Solve the quadratic equation:   n^2 + n - 420 = 0

Factorizing:   (n - 20)(n + 21) = 0

Since n > 0, we get n = 20.

Final result: \boxed{20}

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