Which summation property to use?

Question

I have an answer from a recursion question, for this part I would use a summation property and sub into my general formula, for example if I had

$$ \sum_{i=0}^{n-1}{a^i}. $$ I would just sub in $$ \frac {1-a^n}{1-a}. $$ for the summation.

But I ran into a question where I got the following for my summation: $$ \sum_{i=0}^{k-1}\frac {2^{i-1}}{3^i}. $$

what am I supposed to sub this for? is there a property that can be used here? What should I do if there is no property to replace the summation for a specific question?

Answer 1

There are cases of summation that we do not know how to simplify in general. Fortunately in this case, it is still manageable. $$\sum_{i=0}^{k-1} \frac{2^{i-1}}{3^i}=\frac12 \sum_{i=0}^{k-1} \left( \frac23\right)^i$$

In general, you might want to explore numerical methods if there is no closed form.

Answer 2

$$\sum_{i=0}^{k-1}\frac {2^{i-1}}{3^i}=\sum_{i=0}^{k-1}\frac {2^{i}/2}{3^i}=\frac{1}{2}\sum_{i=0}^{k-1}\left(\frac {2}{3}\right)^i=\frac{1}{2}\frac{1-\left(\frac{2}{3}\right)^k}{1-\frac{2}{3}}=\frac{3}{2}\left(1-\left(\frac{2}{3}\right)^k\right)$$