The first-order difference equationgenerates the sequenceA little simplification shows tha...

The first-order difference equation

generates the sequence

A little simplification shows that the nth term of this sequence is

(a) Show that

(b) Let I represent the annual interest rate, m the number of compounding periods in a year, P0 the initial investment, and d the fixed deposit at the end of each compounding period. Then the balance at the end of each compounding period is generated by the first order difference equation

Use the result of part (a) to show that the balance at the end of n compounding periods is given by