Given the variables of a summation symbol can you find the other variables?

Question

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?

Answer 1

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 $$x^n=x^m-(1-x)T \implies n=\log _x(x^m-(1-x)T)$$

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"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.