Problem 7 · 2010 Math Kangaroo
Easy
Algebra & Patterns
arithmetic-seriesperfect-square
In the picture opposite we see that \(1+3+5+7 = 4\times4\). How big is \(1+3+5+7+\cdots+17+19\)?
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Answer: A — \(10\times10\)
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Hint 1 of 2
The picture shows that adding the first few odd numbers gives a perfect square.
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Hint 2 of 2
Count how many odd numbers you are adding up to 19.
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Approach: sum of first k odd numbers is k squared
- 1+3+5+...+(2k-1) always equals k².
- Here the last term 19 = 2k-1 gives k = 10, so there are ten odd numbers.
- Their sum is 10² = 10x10.
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