Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer South Africa

Pyramids with square bases were built by stacking magnetic balls. A pyramid with a $1 \times 1$ base contains one ball, a pyramid with a $2 \times 2$ base contains five balls, a pyramid with a $3 \times 3$ base contains 14 balls, a pyramid with a $4 \times 4$ base contains 30 balls. How many balls are there in a pyramid with a $10 \times 10$ base?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The $n$ -th pyramid is made by putting the $(n - 1)$ -st pyramid on an $n \times n$ square of balls. Thus the number of balls in the $n$ -th pyramid is $1^2 + 2^2 + \ldots + n^2$ . The number of balls in the 10th pyramid is therefore $1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385$ .

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