How is the following true? Taken from here under ANALYSIS, Large data set. $$ap^1 + ap^2 + ap^3 + ... + ap^k = a\frac{p^{k+1}-1}{p-1}$$ I just can't seem to arrive at the right-hand side. Thanks!
Also please note that this is my first question on here. If this is a stupid question to ask, I'm truly sorry.
On
I believe the formula the text of the question asks us to verify is, in fact, wrongly stated; the correct version is
$a + ap^1 + ap^2 + \ldots ap^k = a \dfrac{p^{k + 1} - 1}{p - 1}, \tag 0$
seen as follows:
Holding the factor $a$ in abeyance for a moment, we have
$(p - 1) \displaystyle \sum_0^k p^i = p \sum_0^k p^i - \sum_0^k p^i$ $= \displaystyle \sum_0^k p^{i + 1} - \sum_0^k p^i = \sum_1^{k + 1} p^i - \sum_0^k p^i$ $= p^{k + 1} + \displaystyle \sum_1^k p^i - \sum_1^k p^i - p^0 = p^{k + 1} - 1; \tag 1$
thus,
$\displaystyle \sum_0^k p^i = \dfrac{p^{k + 1} - 1}{p - 1}; \tag 2$
re-introducing the factor $a$ via multiplication on both sides thus yields
$\displaystyle \sum_0^k a p^i = a\dfrac{p^{k + 1} - 1}{p - 1}, \tag 3$
which is simply another way of writing (1), the desired result.
Hint: $a+ap+ap^2+\ldots=\frac a{1-p}$ for $|p|<1$. And, $$(a+ap+ap^2+\ldots)-(ap^{k+1}+ap^{k+2}+\ldots)=a+ap+\ldots+ap^k$$