Artificial intelligent assistant

Comparing sum of fixed rate value to sum of escalating value Find the number of years, $n$, until the sum of an escalating value/income exceeds the sum of a higher fixed level value/income. Income fixed at £8405.64 Income escalating @ 3% per annum from £5691.84 $$ 8405.64n=5691.84(1+0.03)^n??? $$ Stuck from here, thanks :)

Your equation only uses the year-to-year value of the escalting income and fails to sum it, which is what the problem requires.

Let $x$ be the fixed income, $y$ be the starting income for the escalating case, and $z$ the rate of growth for the latter. Then what you want to find is $n$ such that

$$ x n < \sum_{m=0}^{n-1}yz^m = y \sum_{m=0}^{n-1}z^m $$

The term $\sum z^m$ is a geometric series, for which there exist known formulas for its value. Specifically, see here.

**EDIT** :

To solve the above, you need to find the value of $n$ such that

$$ y {{1-z^n}\over{1-z}} - xn > 0, $$

which I don't think has an analytic solution. Since $n$ can only take integer values, it's probably easiest to just to plug in $n=1,2,\dots$ until the above difference become positive. Another option is to solve graphically. I get the following:

!enter image description here

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