Problem 16 · 2010 Math Kangaroo
Hard
Arithmetic & Operations
arithmetic-seriesperfect-square
The picture shows that 1 + 3 + 5 + 7 = 4 × 4. How big is 1 + 3 + 5 + 7 + … + 17 + 19?
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Answer: A — \(10 \times 10\)
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Hint 1 of 2
The picture shows a pattern: adding the first few odd numbers makes a perfect square.
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Hint 2 of 2
Count how many odd numbers are in the list \(1, 3, 5, \ldots, 19\); that count, squared, is the answer.
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Approach: odd numbers build a square
- The picture shows \(1 + 3 + 5 + 7 = 4 \times 4\): four odd numbers make a \(4 \times 4\) square.
- The list \(1, 3, 5, \ldots, 19\) has 10 odd numbers, so it builds a \(10 \times 10\) square.
- So the sum is \(10 \times 10\) — the answer is A.
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