Describe the relationship between the mathematics of calculating compound interest, and the mathematics behind analysing geometric sequences
How might compound interest be described as a problem of applied geometric sequences?
A geometric sequence is a sequence where each term is determined by multiplying the previous term of the sequence by a constant value. The nth term of a geometric sequence is given by `T_n = a*r^(n-1)` where a is the first term of the series and n is the common ratio.
In the determination of compound ratio it has to be kept in mind that interest is being calculated not on the initial amount but on the initial amount as well as the interest earned till then.
This gives the interest earned after n time periods as `I = P(1 + i)^n - P` where P is the initial principle and i is the rate of interest for a time period. The formula for calculating compound interest is a geometric series with the first term of the series equal to the initial principle and the common ratio being 1 + i.