Geometric series with modified terms

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SUMMARY

The discussion focuses on summing a modified geometric series represented by the formula \(\sum\limits_{i=0}^N p^{i} q^{ti}\), where \(t\) is a constant. The user seeks assistance in deriving a solution for this series, building on the standard geometric series formula \(\sum\limits_{i=0}^N p^{i} = \frac{1-p^{N+1}}{1-p}\). The transformation of the series into the form \(P = p \cdot q^{t}\) is also highlighted as a key step in the process.

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  • Understanding of geometric series and their summation
  • Familiarity with mathematical notation and series manipulation
  • Knowledge of constants and variables in mathematical expressions
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kop442000
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I am looking for a way to sum some numbers. I understand that if I want to sum pi, I can use the geometric series:

[itex]\sum\limits_{i=0}^N p^{i} = \frac{1-p^{N+1}}{1-p}[/itex]

But can anyone help me with what to do when I need:

[itex]\sum\limits_{i=0}^N p^{i} q^{ti}[/itex]

where t is just a constant.

Thank you in advance of any help!
 
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Let : [itex]P = p.q^{t}[/itex]
 

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