Observe the pattern: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9. Using this pattern, find the sum of the first 8 odd numbers and explain why the pattern works.
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Step-by-Step Solution
Step 1: Analyze the given pattern
We are given a pattern where the sum of the first n odd numbers is equal to n2. Let's observe the first few terms to confirm this. The first odd number is 1, and its sum is 1=12. The sum of the first two odd numbers (1+3) is 4=22. The sum of the first three odd numbers (1+3+5) is 9=32.
Step 2: Generalize the pattern
From the observed pattern, we can generalize that the sum of the first n odd numbers is equal to n squared. This is a known property of arithmetic progressions.
Step 3: Calculate the sum of the first 8 odd numbers
Using the generalized pattern, to find the sum of the first 8 odd numbers, we simply need to calculate 82. This gives us 64.
Step 4: Explain why the pattern works (Visual Proof)
This pattern works because the sum of consecutive odd numbers can be visualized as building up a square. Start with a 1×1 square (1). To get a 2×2 square (4), you add 3 units in an 'L' shape around the 1×1 square. To get a 3×3 square (9), you add 5 units in an 'L' shape around the 2×2 square. Each time, you add the next odd number to form the next larger square.
Step 5: Explain why the pattern works (Arithmetic Progression)
The sequence of odd numbers (1,3,5,…) is an arithmetic progression with the first term a=1 and common difference d=2. The sum of the first n terms of an arithmetic progression is given by the formula Sn=2n[2a+(n−1)d]. Substituting a=1 and d=2 into the formula, we get Sn=2n[2(1)+(n−1)2]=2n[2+2n−2]=2n[2n]=n2. This proves the pattern mathematically.