# Geometric Series..

You are offer $2$ lucrative investment schemes. The $1$st scheme awards £$4000$ on the 1st day and every subsequent day this amount is increased by £$8000$. The $2$nd scheme awards £$0.01£ on the first day and everyday this amount doubles. How many days does it take for the$2$nd scheme to become more than the first scheme. I've worked out in my head and by calculator that it takes 26days however how would i solve this using a mathematical method ## 2 Answers You have on the$n$th day$8000n-4000$pounds in the first scheme and$2^{n-1}/100$in the second. If you set these equal, and then find the next integer above the solution, then you will have the answer. You might be able solve this using something like the Lambert W function but you will find it easier to use numerical methods, such as the one you found. If you wanted the cumulative sum to be greater, you would be comparing$4000(n^2+n-1)$with$(2^n - 1)/100$and get an answer of$29$days. You write down an expression that calculates the amount you have with each scheme after$n$days. The first is an arithmetic series. If the first day is day$0$, on day$n$you get$4000+8000n$and up to day$n$you have received$\sum_{i=0}^n (4000+8000i)$For the second, which is a geometric series, you get$0.01\cdot 2^n$on day$n$, and up to day$n$you get$\sum_{i=0}^n 0.01\cdot 2^i$Sum up each of these, set them equal, solve for$n\$. You will need a numeric solution.

• zian probably did not add them up Commented Feb 27, 2014 at 23:32
• @ross i fully understand that but how do i solve that for n?
– zian
Commented Feb 27, 2014 at 23:33
• Equations with polynomials and exponentials don't yield to algebraic solution most of the time, including this one. You will have to use numerics Commented Feb 27, 2014 at 23:45