Lesson 1

Recurrence relations often arise in finance — when calculating the income from a loan, the debt from a mortgage, or the amortized cost of a product under insurance. Very often, in such calculations, the value \(V(n)\) of some monetary characteristic through \(n\) time intervals is given not just as a formula of \(n\), but as an expression of \(V(n-1)\).

Suppose, for example, that you start investing with 100\(\$\) with an annual interest rate \(5\%\). Each year you add 10\(\$\). Thus, the amount \(V(n)\) in your account after \(n\) years satisfies the following recurrence relation: \[V(0)=100, \ V(n)=1.05\cdot V(n-1)+10 \ .\]

Problem. Calculate \(\lfloor V(10) \rfloor\).