The finite geometric series summation formula for \(\sum_{k=0}^{n - 1} ar^{4k}\) can be derived using the standard formula for geometric series. Specifically, the formula \(\sum_{k=0}^{n - 1} ar^{k} = a\frac{1 - r^n}{1 - r}\) can be adapted by substituting \(r\) with \(r^4\). This results in the formula \(\sum_{k=0}^{n - 1} ar^{4k} = a\frac{1 - (r^4)^n}{1 - r^4}\). The discussion highlights the simplicity of this adaptation, as noted by user micromass.
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micromass said:Use the formula with [itex]r^4[/itex] instead of r.