3
$\begingroup$

I need to turn the following into something that doesn't use summation notation. Can someone help me figure out how to do that? I would know how to do it were it a simpler case but this one is difficult.

$\sum_{i=0}^{\log n}5^{\log n-i}\left(3\cdot2^i+2\right)$

My log's here are base 2.

  • 0
    Please$ $make$ $sure$ $that I correctly interpreted your expression.2012-10-28

1 Answers 1

1

$\begin{align*} \sum_{i=0}^{\log n}5^{\log n-i}\left(3\cdot2^i+2\right)&=5^{\log n}\sum_{i=0}^{\log n}\left(3\left(\frac25\right)^i+2\left(\frac15\right)^i\right)\\ &=5^{\log n}\left(3\sum_{i=0}^{\log n}\left(\frac25\right)^i+2\sum_{i=0}^{\log n}\left(\frac15\right)^i\right)\;. \end{align*}$

The last two summations are simple geometric series. Assuming that the upper limit of summation is actually $m=\lfloor\log n\rfloor$, they are

$\sum_{i=0}^m\left(\frac25\right)^i=\frac{1-\left(\frac25\right)^{m+1}}{1-\frac25}=\frac53\left(1-\left(\frac25\right)^{m+1}\right)$

and

$\sum_{i=0}^m\left(\frac15\right)^i=\frac{1-\left(\frac15\right)^{m+1}}{1-\frac15}=\frac54\left(1-\left(\frac15\right)^{m+1}\right)\;.$

  • 0
    @Jon: You’re welcome.2012-10-29