Question

(7) An employee joined a company in 2017 with a starting salary of 50000 USD. Every year this employee receives a raise of 1000 USD plus 5 percent of the salary of the previous year. Let Cn denote the employee's salary n years after 2017.
(a) Set up a recurrence relation for Cn.
(b) What will the salary of this employee be in 2025?
(c) Find an explicit formula for Cn.

Answer 1

C₀=50 000

a.)C_n=C_n-1*1.05 + 1 000

b.)C₁=53 500

C₂=57 175

C₃=61 033.75

C₄₌65 085.4375

C₅=69 339.7094

C₆=73 806.6949

C₇=78 497.0296

C_2025-2017=C₈=83 421.8811  

c.) $C=50000$

$k= 1 + 5/100$

$d=1000$

$C$₁$=C*k+d,$

$C$₂$=C$₁$*k+d=C*k^2+d*k+d$ $=$

$C*k^2+d*(k+1),$

$C$₃$=C*k^3+d*(k^2+k+1),$

$C_4$ $=C*k^4+d*(k^3+k^2+k+1),$

$\sum_{i=0}^{n-1} k^i=(k^n-1)/(k-1),$

C_n$=C*k^n+d*(k^n-1)/(k-1)=50000*1.05^n+1000*(1.05^n-1)/0.05$