a little bit confused about summation rules

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a little bit confused about summation rules

Mathematics
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Try reading this: http://psychassessment.com.au/PDF/Chapter%2001.pdf
ok so I have this formula \[v_{av}=\frac{\sum^n_{i=1}v_i\delta t_i}{\sum^n_{i=1}\delta t_i}\]
yeah I have looked it up. I would just like some clarification on a problem I am working on

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I also have that \[\delta t_i=\frac{2z_i}{v_i}\]
so plugging this in I get \[v_{av}=\frac{\sum^n_{i=1}v_i\frac{2z_i}{v_i}}{\sum^n_{i=1}\frac{2z_i}{v_i}}\]factoring out constants\[v_{av}=\frac{\sum^n_{i=1}v_i\frac{z_i}{v_i}}{\sum^n_{i=1}\frac{z_i}{v_i}}\]canceling\[v_{av}=\frac{\sum^n_{i=1}z_i}{\sum^n_{i=1}\frac{z_i}{v_i}}\]
now I am wondering if I can simplify this any further?
\(v_i\) and \(z_i\) are all different terms depending on index, so I can't see how you could simplify that. I could be wrong...
right that's what my thinking was.
I mean, what about this?\[v_{av}=\frac{\sum^n_{i=1}\cancel v_i\frac{z_i}{\cancel v_i}}{\sum^n_{i=1}\frac{z_i}{v_i}}\]or am I misreading your notation?
\[\sum_i(v_i)\cdot(\frac{z_i}{v_i})=\sum_iz_i\]is valid
as long as it's under the same summation sign
hmm no I don't think that is correct. This is a physics problem and the answer does not make sense. v is a velocity and z is a distance so the final sum is a velocity.
or final answer I should say.
you are right, that rule wouldn't make sense mathematically either
actually the more I thinking about it the more I am thinking it should make sense, but I am obviously unsure... @Zarkon @satellite73 series rules question
this is valid... \[\sum_i\left[(v_i)\cdot\left(\frac{z_i}{v_i}\right)\right]=\sum_iz_i\]
right and I did that in the numerator of the fraction to get. \[v_{av}=\frac{\sum^n_{i=1}z_i}{\sum^n_{i=1}\frac{z_i}{v_i}}\] Is there any way to express this as a product of \(z_i\) and \(v_i\) in the numerator?
unless you have more info...you can't simplify it any more
thanks

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