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2 | Page 3 | Proof(s)
Example 4--Page 2
So what is the second way? You should notice that the only thing that is common on both sides is that they both have , and this is our key to prove this identity by the second way. Usually we keep things that already are the same on both sides. So we keep , and try to combine all other terms. And if we want to reduce the differences of operations, which means change addition to multiplication, we need to become a common factor, so later we can factor the left side.
Now, how can we get ? We see a lot of 2s at left side, like , and there is no argument that we should combine either , or , or , and one of these should work out this problem. We chose , because which gives .
Then what do we do to ? We don't have a formula, but we have a Sum formula of right? Here is what we're going to get:
We have a little problem here, the last term does not have ! That's OK, we believe you can figure out that .
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