I don't know the technical language for what I'm asking, so the title might be a little misleading, but hopefully I can convey my purpose to you just as well without.
Essentially I'm thinking of this: the series 4n+4n−1⋯4n−n.
I suppose this is the summation of the series 4n from n to 0.
But is there any way to express this as a pure equation, not as a summation of a series?
If so, how do you figure out how to convert it?
Answer
In general, for x≠1 it is true that
n∑k=0xk=1+x+⋯+xn=xn+1−1x−1.
So, in your case in particular, we have that
n∑k=04n−k=4n+⋯+4+1=1+4+⋯+4n=n∑k=04k=4n+1−13.
Alternatively, one could pull out a factor of 4n from all terms, and compute
n∑k=04n−k=4nn∑k=0(14)k=4n⋅(14)n+1−1(14)−1=4n⋅4n+1−14n+134=4n+1⋅4n+1−14n+13=4n+1−13.
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