3
$\begingroup$

$ 3= \sum_{n=1}^{t} \frac{1}{1.08^n} $

I see that it is $3 = 1.08^{-t}(12.5 \times 1.08^t{-12.5})$ (from Wolfram Alpha, but I'm not sure how to get it. I tried solving as a geometric series, I had problems and didn't get the correct answer.

I see that $ t\approx3.56592$, which seems like it's correct, but I have no idea where the 12.5 and all that stuff came from. Unfortunately my calculus book doesn't help much, as it is mainly focus on infinite series.

  • 0
    Is this homework?2010-10-06
  • 1
    Yes, but it's the end of a problem for a Financial Mathematics course dealing with internal rate of return. I can write that the answer is 3.56592, but I want to know how to solve this problem.2010-10-06
  • 2
    $\sum_{n=1}^{t}$ does not make much sense when $t$ is fractional. Perhaps the question is phrased differently?2010-10-06
  • 1
    @Moron: The sum $S$ is a function of the number of terms, but we can "continue" $S(t)$ to the reals.2010-10-07
  • 0
    @Americo: Yes, but how is that relevant?2010-10-07
  • 1
    @Moron: In my opinion, only the context establishes its relevancy or not. If one is asked to interpolate between number of capitalization periods (e.g. to analyze a scenario with different interest rates) it makes sense. Or when one studies the sensitivity of $S(t)$ to deviation from the estimated life ($t$ periods, months, years,etc.). See e.g. chapter on *Sensitivity Analysis* in *Engineering Economics* by Riggs, Bedforth and Randhava.2010-10-07
  • 0
    @Americo: Exactly! The context is relevant, that is why I was asking OP for a different phrasing of the problem which will make it clearer (see my earlier comments). As the question is currently stated the only context I can meaningfully derive is that t is a natural number, and the fact that S(t) can be extended to reals/complex is kind of irrelevant... You might have interpreted it different, but I find that it is better to let OP clarify rather than we try to guess what OP might have meant.2010-10-07

3 Answers 3