Hints and solutions to Look for a pattern #1

What is the sum of the first 4 odd numbers? of the first 5 odd numbers? of the first 7 odd numbers? of the first 10 odd numbers? of the first 2356 odd numbers?

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• The first few of these are easy -- 1 + 3 + 5 + 7 = 16, 1 + 3 + 5 + 7 + 9 = 25 - and you may begin to notice the pattern that the sum is always a square number. In fact, it is the square of the number of odd numbers you add together. So the sum of the first 7 odd numbers is 7² = 49, and the sum of the first 10 odd numbers is 100. Using a calculator or long multiplication, you can find that the sum of the first 2356 odd numbers is 2356² = 5550736.
• The figure on the problem page shows that each odd number adds an additional row and column to the square. And the area of the square, n², is the sum of the first n odd numbers.