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I am having a bit of trouble with this passage in the summation, could you guys help understand what's happening?

$$\frac{1}{4}\sum_{i\leq{n}}({\frac{2}{3}})^n=\frac{3}{4}(\frac{2}{3})^i$$

Is it a purely algebraic passage or is it something more related to the problem I am currently doing

Thank you very much in advance!

Ramiro genta
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    The way in which this expression is written seems very unusual. It is unclear if this was meant to be with $i$ as the indexing variable for the summation or if $n$ is. If it was $i$ as the indexing variable (which is the more common since it appears first in the condition) then it is strange that $n$ appears in the expression for the summands and stranger still that the final result on the right hand side involves $i$ and not $n$. This implies to me that $n$ would have been the indexing variable... If that were the case then write as $\frac14\sum\limits_{n=i}^\infty\left(\frac23\right)^n$ – JMoravitz Jul 20 '22 at 12:46

2 Answers2

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You can rewrite the sum as $$\sum_{n\geq i}\left(\frac{2}{3}\right)^n=\sum_{n=0}^{+\infty}\left(\frac{2}{3}\right)^n-\sum_{n=0}^{i-1}\left(\frac{2}{3}\right)^n$$

Hamdiken
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It’s the sum to infinity of a geometric series the first term of which is $\left(\frac23\right)^i$ and common ratio $\frac 23$

David Quinn
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