Question

Suppose that an initial bank deposit was $2.000 At the end of each year it is increased by 5% and an additional $500 is deposited amount of the deposit after the nth year. Write the recurrence relation with initial condition for the amount deposited after nth year. Next provide the solution.

Answer 1

Please find the solution with explanation:
Initial condition
P0 = 2000(initial deposit)

At the end of first year
P1  = principal + interest on principal + additional deposit
P1  = 2000 + ((5/100)* 2000) + 500
    = 2000 + 100 + 500
    = 2600

At the end of second year
P2  = previous principal(P1) + interest on previous principal(P1) + additional deposit
P1  = 2600 + ((5/100)* 2600) + 500
    = 2600 + 130 + 500
    = 3230
...
...
...
Pn  = previous principal(Pn-1) + interest on previous principal + additional deposit
Pn  = Pn-1 + ((5/100)* Pn-1) + 500
    = Pn-1 + 0.05Pn-1 + 500
    = 1.05Pn-1 + 500
    
Solution:
P0 = 2000 (initial condition)
Pn = 1.05Pn-1 + 500 (Recurrance relation)