Basic definition of a sum
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Where :
Where :
Shifting of starting and ending indices
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Proof of the equality of the shifting of indices:
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Smaller summation notation
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Combinations proof (used in below proof)
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Proof by mathematical induction of the recursive geometric series (uses recursive summation notation)
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Definition
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Base case (and some specific examples)
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Inductive step
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Shifting of starting and ending indices (see above for proof):
See combinations proof above:
Shifting of starting and ending indices (see above for proof):
Adding case k=0 to the summation, means that the same must be subtracted from the summation:
Terms cancel out.
Q.E.D.
A general formula for recursive summation series
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First proof, used in second proof
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One method
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Inductive method
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Second proof, this one for the general formula for recursive summation series
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Miscellaneous items (some valid, some not)
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これ、ちょっとちがうね。
これもちがう。