0+2+4+⋯+(2n−2)=n(n−1) for all n ∈Z+
What I have:
Let n exist in Z^+. If n = 2, then L.H.S. = 2, and the R.H.S. = 2(2-1) = 2. So, L.H.S. = R.H.S. and this holds for n = 2. Assume this holds for some k existing in Z^+. That is 0 + 2 + 4 + ... + 2(k-2) = k(k-1)
What I am stuck on:
How do I show that this holds for 0 + 2 ... (2k-2) = k(k-1)?
Answer
You assume that for n=k, the statement holds:
0+2+⋯+2(k−1)=k(k−1).
You then want to prove the statement for n=k+1, that is, we want to show
0+2+⋯+2(k−1)+2k=(k+1)k.
Can you make the connection from (a) to (b)?
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