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Relate square numbers to sums of consecutive odd numbers, triangular numbers, and recognize patterns in successive differences.
Adding consecutive odd numbers yields square numbers.
Did you know that perfect squares are deeply connected to odd numbers? Let's explore the interesting pattern that connects them.
Using the nth odd number rule to find the next square.
Problem. Find , given that .
Successive subtraction of odd numbers.
Problem. Determine if and are perfect squares by using successive subtraction.
Check if 81 is a perfect square by subtraction.
We can find the square root of a perfect square by successively subtracting consecutive odd numbers starting from 1. Let us consider finding the square root of 81 using this method. We begin by subtracting the first odd numbers: 81 - 1 = 80, 80 - 3 = 77, and 77 - 5 = 72. Continuing this pattern, we get 72 - 7 = 65, 65 - 9 = 56, 56 - 11 = 45, and 45 - 13 = 32. The next procedural step is to subtract 15, which gives 32 - 15 = . Finally, we subtract the next consecutive odd number to reach zero, calculated as 17 - = 0. Since we successively subtracted odd numbers until we obtained 0 at the step, the square root of 81 is 9.
Order the steps for successive subtraction.
Drag the steps into the correct order to verify that 36 is a perfect square using successive subtraction.
Connecting triangular numbers to squares.