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For example, $$T= \sum_{i=0}^n x^i$$ Given the values of T, x, and where i begins, can a mathematical equation be written to find the value of n. If so, can you find the value of any variable in this summation given information of the other variables?

Mike Wilson
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1 Answers1

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Suppose $T, x,$ and the starting value $i$ are given (say, $m$). Then you can, indeed, find the value of $n$ as follows:

The case $x=1$ trivial. $$T=\sum_{i=m}^n1=n-m$$ Now assume $x\not =1:$ $$T=\sum_{i=m}^nx^i=\underbrace{x^m+x^{m+1}+\cdots x^n}_{\text{a geometric progression}}=\frac{x^m(1-x^{n-m})}{1-x}$$ From here $$x^n=x^m-(1-x)T \implies n=\log _x(x^m-(1-x)T)$$

"If so, can you find the value of any variable in this summation given information of the other variables?"

The answer is "yes". In fact you can solve the above equation $$x^n=x^m-(1-x)T$$ for $m,T$ or $x$, as well.

VIVID
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