Answer:
a) It will take 17.71 years
b) It will take 17.58 years
c) I will earn $6.60 more in compound continuously
Step-by-step explanation:
a) Lets talk about the compound interest
- The formula for compound interest is A = P (1 + r/n)^(nt)
, Where:
- A = the future value of the investment, including interest
- P = the principal investment amount (the initial deposit)
- r = the annual interest rate (decimal)
- n = the number of times that interest is compounded per unit t
- t = the time the money is invested
* Lets solve the problem
∵ The money deposit is $2000
∵ The rate is 6.25%
∵ The interest is compound quarterly
∵ The future value is $6000
∴ P = 2000
∴ A = 6000
∴ r = 6.25/100 = 0.0625
∴ n = 4
∴ t = ?
∵ A = P (1 + r/n)^(nt)
∴ 6000 = 2000 (1 + 0.0625/4)^4t ⇒ divide both sides by 2000
∴ 3 = (1.015625)^4t ⇒ insert ㏑ for both sides
∴ ㏑(3) = ㏑(1.015625)^4t
∵ ㏑(a)^b = b ㏑(a)
∴ ㏑(3) = 4t ㏑(1.015625) ⇒ divide both sides by ㏑(1.015625)
∴ 4t = ㏑(3)/㏑(1.015625) ⇒ divide both sides by 4
∴ t = [㏑(3)/㏑(1.015625)] ÷ 4 = 17.71
* It will take 17.71 years
b) Lets talk about the compound continuous interest
- Compound continuous interest can be calculated using the formula:
A = P e^rt
- A = the future value of the investment, including interest
- P = the principal investment amount (the initial amount)
- r = the interest rate
- t = the time the money is invested
* Lets solve the problem
∵ The money deposit is $2000
∵ The rate is 6.25%
∵ The interest is compound continuously
∵ The future value is $6000
∴ P = 2000
∴ A = 6000
∴ r = 6.25/100 = 0.0625
∴ t = ?
∵ A = P e^rt
∴ 6000 = 2000 e^(0.0625 t) ⇒ divide both sides by 2000
∴ 3 = e^(0.0625 t) ⇒ insert ㏑ to both sides
∴ ㏑(3) = ㏑[e^0.0625 t]
∵ ㏑(e^a) = a ㏑(e) ⇒ ㏑(e) = 1 , then ㏑(e^a) = a
∴ ㏑(3) = 0.0625 t ⇒ divide both sides by 0.0625
∴ t = ㏑(3)/0.0625 = 17.5778
* It will take 17.58 years
c) If t = 5 years
# The compound quarterly:
∵ A = P (1 + r/n)^(nt)
∴ A = 2000 (1 + 0.0625/4)^(4×5)
∴ A = 2000 (1.015625)^20 = $2727.08
# Compound continuously
∵ A = P e^(rt)
∴ A = 2000 e^(0.0625×5) = $2733.68
∴ I will earn = 2733.68 - 2727.08 = $6.60
* I will earn $6.60 more in compound continuously
, so the series converges (and thereby has a sum), so the answer is D.