Problem 11 · AMC 8 Stretch
Core
Number Theory
Geometry & Measurement
visual-representationpattern-recognition
Look at these sums: \(1 = 1\), \(1+3 = 4\), \(1+3+5 = 9\), \(1+3+5+7 = 16\). Adding up the first \(n\) odd numbers always gives a perfect square. What is the sum of the first \(10\) odd numbers?
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Answer: 100
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Hint 1 of 4
First just look at the answers: \(1, 4, 9, 16, 25, \dots\) What kind of numbers are these?
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Hint 2 of 4
Draw each sum as dots arranged in a square. Start with \(1\) dot. How do you grow the square to the next size?
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Hint 3 of 4
To turn a \(k \times k\) square into a \((k+1) \times (k+1)\) square, you add an L-shaped border: \(k\) dots down the new right side, \(k\) dots along the new bottom, and \(1\) dot in the corner.
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Approach: Picture proof — each odd number is an L-shaped square border
- The answers \(1, 4, 9, 16, 25, \dots\) are the perfect squares \(1^2, 2^2, 3^2, 4^2, \dots\)
- Build the sum as a square of dots. To grow a \(k \times k\) square into the next square, add an L-shaped border: a column of \(k\) dots, a row of \(k\) dots, and \(1\) corner dot.
- That border has \(k + k + 1 = 2k + 1\) dots — exactly the next odd number. So \(1 + 3 + 5 + \dots + (2n-1) = n^2\).
- Adding the first \(10\) odd numbers therefore gives \(10^2 = 100\).
Mark:
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