sequences and series problem - returns on investment

Question

I am having a problem with these types of problem.

Kenny is offered 2 investment plans , each requiring an initial investment of £10,000.
Plan A offers a fixed return of £800 per year. - arithmetic sequence Un=a+(n-1)d Plan B offers a return of 5% each year, reinvested in the plan. - geometric sequence Un=ar^(n-1) over what period of time is plan A better than plan B?

10000+(n-1)800 > 1000(1.05)^(n-1)

When I try to solve it I get to the point where the best I can do is try different values of n. There's no way for me to solve it with n's on both sides and I can't figure out a way to isolate it. Unless I got my equations wrong. tia

Answer 1

Hint

You are first looking for the sero of function $$f(n)=10000+800(n-1)-1000\left(\frac{105}{100}\right)^{n-1}$$ There is an analytical solution which involves non-elementary functions.

If you made a graph, this fuction varies very fast. So, to make it easier, take logarithms and then consider instead the function $$g(n)=\log\big[10000+800(n-1)\big]-(n-1) \log\left(\frac{105}{100}\right)-\log(1000)$$

Plot $g(n)$ and you will see very easily the solution.

Answer 2

There is no way to solve this equation with ordinary algebra. Guess and adjust your guess is what you have to do.

Nice work.