A Registered Education Savings Plan (RESP) earns interest at a rate of 5% per year, compounded annually. Jasmine’s parents invest $4000 in the account today.
a) Determine an explicit formula to represent the value of the investment. (2 marks)
b) Use your formula to write the first four terms of the sequence. (4 marks)
c) What will the investment be worth in
i) 9 years? (1 mark)
ii) 16 years? (1 mark)
d) Approximately how long will it take for the investment to double? (2 marks)
a) The explicit formula to represent the value of the investment in an RESP is given by:
V = P(1 + r)^n
Where:
V = Value of the investment
P = Principal amount (initial investment)
r = Annual interest rate (as a decimal)
n = Number of years
In this case, the principal amount P is $4000 and the annual interest rate r is 5% (or 0.05 as a decimal). Therefore, the explicit formula for the value of the investment in Jasmine's RESP is:
V = 4000(1 + 0.05)^n
b) To write the first four terms of the sequence, we substitute different values for n into the formula:
When n = 0:
V = 4000(1 + 0.05)^0 = 4000(1) = 4000
When n = 1:
V = 4000(1 + 0.05)^1 = 4000(1.05) = 4200
When n = 2:
V = 4000(1 + 0.05)^2 = 4000(1.05^2) ≈ 4410
When n = 3:
V = 4000(1 + 0.05)^3 = 4000(1.05^3) ≈ 4631.25
So, the first four terms of the sequence representing the value of the investment are $4000, $4200, $4410, and approximately $4631.25.
c) i) To determine the value of the investment after 9 years, we substitute n = 9 into the formula:
V = 4000(1 + 0.05)^9 ≈ $6381.63
Therefore, the investment will be worth approximately $6381.63 after 9 years.
ii) To determine the value of the investment after 16 years, we substitute n = 16 into the formula:
V = 4000(1 + 0.05)^16 ≈ $8888.25
Therefore, the investment will be worth approximately $8888.25 after 16 years.
d) To determine approximately how long it will take for the investment to double, we need to find the value of n when V becomes $8000 (double of the principal amount). So, we set up the following equation:
8000 = 4000(1 + 0.05)^n
Dividing both sides by 4000, we get:
2 = (1.05)^n
Taking the logarithm (base 1.05) of both sides:
log(2) = nlog(1.05)
Using logarithm properties, we can solve for n:
n ≈ log(2) / log(1.05) ≈ 14.21
Therefore, it will take approximately 14.21 years for the investment to double.
balance = 4000 (1.05)^n
geometric sequence a = 4000, r = 1.05
4000 4200 4410 etc