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: