Question

3b) A bank pays you 4.5% interest per year. In addition, you receive |100 as bonus at

the end of the year (after the interest is paid). Find a recurrence for the amount of

money after n years if you invest |2000.

Answer 1

Let r=0.045 is an interest rate, S₀=2000 the initial amount of money, Sn - the amount of money after n years, and b=100 is the annual bonus. Then we have a recurrent equation:

$S_n=(1+r)S_{n-1}+b$

$S_n-S_{n-1}=r(S_{n-1}+b/r)$

$(S_{n}+b/r)-(S_{n-1}+b/r)=r(S_{n-1}+b/r)$

$S_{n}+b/r=(1+r)(S_{n-1}+b/r)$

The last equation shows that the sequence $S_{n}+b/r$ is a geometric progression, and we can write

$S_{n}+b/r=(1+r)^n(S_{0}+b/r)$

$S_{n}=(1+r)^n(S_{0}+b/r)-b/r$

$S_{n}=(1+0.045)^n(2000+100/0.045)-100/0.045=4222.22\cdot1.045^n-2222.22$

Answer. $S_{n}=4222.22\cdot1.045^n-2222.22$