Exercise: Let a and b be real numbers and let n be a natural number. Verify that (a - b)^2 = (a^2 - 2ab + b^2). (Hint: Start with the right-hand side and show that it can be simplified to obtain the left-hand side.) Verify that (a - b)^2 = a^2 - 2ab + b^2. Use part (b) to reprove part (a) in summation notation. Also, verify that (a - b)^2 = a^2 - 2ab + b^2.
Added by Joseph J.
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We can start by expanding the right-hand side: $$(6-a)(6^{n-1} + 6^{n-2}a + \cdots + a^{n-1}) = 6^n - a^n$$ We can expand the right-hand side using the distributive property: $$6^n - 6^{n-1}a + 6^{n-1}a - 6^{n-2}a^2 + \cdots - a^n = 6^n - a^n$$ Now, we can see Show more…
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