At the end of each year

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

At the end of each year, a salesperson is paid a bonus of

2000

, which is always deposited into the same account. It earns a fixed interest rate of

6\%

per annum, with interest paid annually. The total amount in the account at the end of each year is given by

A_1=2000

A_2=A_1\times1.06+2000

A_3=A_2\times1.06+2000\text{, and so on.}

(1) Show that

A_3=2000+2000\times1.06+2000\times(1.06)^2

. (2) Show that

A_4=2000[1+1.06+(1.06)^2+(1.06)^3]

. (3) Find the total bank balance after

10

years, assuming there are no fees or withdrawals.