62 years old means let t = 62
You want $1 000 000 at the end so y(t) = 1 000 000

must use y(t) = Ce^kt? continuously means daily?

We can let the one-time deposit be x.

After the first compounding period, the total amount = 1.05x (because you're adding 5% of the principal amount and 5% = 0.05.


If compounded yearly,

At the end of 62 years, the amount of money will be = (1.05)⁶²x

(Basically, compounding means the total amount after any compounding period is always 1.05 times of the previous period))


Since they want to reach a total of $1 000 000,

Then (1.05)⁶²x = 1 000 000

x = 1 000 000 / (1.05)⁶²

x ≈ 48 558.2982
x = 48 558.30 (nearest cent)

interesting ..... I am very keen to know what is the answer given by the school...

However, if they want continuous compounding, then we are looking at an infinite number of compounding periods.

https://www.investopedia.com/terms/c/continuouscompounding.asp

The formula will be :

Total = P × (1 + i/n)ⁿᵗ

Whereby

P is the one time deposit (principal sum)
i is the interest rate
n is the number of compounding periods in 1 year
t is the number of years


Since we want an infinite number of periods, n tends to infinity (n → ∞)

When this occurs, the limit of (1 + i/n)ⁿᵗ is :

eⁱᵗ


This is actually the basis of the how e is derived (you can read up on it online)

yes. I know the theory. we learnt this under radioactively decay whereby dN/dt = -kN.. then solve this DE to get N = No e^kt . where the two k never change value.... Applying here dN/dt is the interest obtained per dt period of time. and k is the growth rate per dt of time too. dt is infinitely small.... So if we used it here, dt is in second? or smaller? To me, using this equation has assumed continuity. Using the commercial equation, we are considering the interest is credited yearly. Very different... That's y my answer is different.

So, Total = Peⁱᵗ

In the context of your question,

y(t) = Ceᵏᵗ

1 000 000 = Ce⁰·⁰⁵⁽⁶²⁾

Your total amount y(t) = $ 1 000 000
Principal amount (the one-time deposit) = C
k is the interest rate (5% = 0.05 , for each period compounded)
t = 62 (62 years)


C = 1 000 000 / e⁰·⁰⁵⁽⁶²⁾

C = 1 000 000 / e³·¹

C ≈ 45 049.2024

C = 45 049.20 (2d.p for nearest cent)

@Yong Kc :

Yes agree. I guess that's why Zwen said the setter is cunning.

They have to turn it into a DE or be able to interpret that y(t) = Ceᵏᵗ in the correct way.

The 6 marks is actually easy to get if they simply substitute the values accordingly.

Thank you all for the explanation and workings. Appreciate it. By the way, the answer key is $45,049.20.

Welcome. For further reading :


https://betterexplained.com/articles/definitions-of-e-colorized/

https://www.mathsisfun.com/money/compound-interest-periodic.html

Alright thanks