Problem 6 · 2025 Math Kangaroo
Medium
Algebra & Patterns
sequence-of-figuresperfect-square
The pictures shown are the first three pictures in a sequence. How many dots does the fifth picture in the sequence consist of?
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Answer: A — 72
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Hint 1 of 2
Count the dots in the first three diamonds, then look at how the totals grow.
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Hint 2 of 2
The counts are 8, 18, 32… which are 2×2², 2×3², 2×4², so picture k has 2(k+1)² dots.
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Approach: find the closed form of the count
- Pictures 1, 2, 3 have 8, 18, 32 dots = 2·2², 2·3², 2·4².
- Picture k has 2(k+1)² dots.
- Picture 5 has 2·6² = 72.
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