$2\,000\,000 = \frac{1}{1.1^5} \cdot \sum_{t=1}^{10} \frac{e}{1.7^t}$

Question

Let

$$2\,000\,000 = \frac{1}{1.1^5} \cdot \sum_{t=1}^{10} \frac{e}{1.7^t}$$

($e$ is not Euler’s number here)

Apparently $e$ is $399\,382.63$ but how can one find that out?

If I multiply $20\,000\,000$ with $1.1^5$ I get $3\,221\,020$. But then? I can't multiply with $1.07^t$ because it's in the sum.

Answer 1

Your equation is of the form $a=eb \sum_{t=1}^{10}c^t.$

We have $\sum_{t=1}^{10}c^t=c \frac{1-c^{10}}{1-c}.$

Thus

$$a=ebc \frac{1-c^{10}}{1-c}.$$

$a,b$ and $c$ are given, hence

$$e=\frac{a}{bc \frac{1-c^{10}}{1-c}}.$$