A man invests an amount, A, into a bank account. After 15 years (in the 16th year) he deposits another $10 000.
After 50 years he has (in the 51st year) $251 894.21 in the account. If the amount of money in the account doubles
every 15 years, how much was his initial deposit A? [4]

The initial amount, A is approximately $19,991.1.

Step-by-step explanation:

The initial amount the man invests in the bank = A

The amount the man deposits after 15 years = $10,000

The amount in the account after 50 years = $251,894.21

The amount of money after every 15 years = 2 × Initial amount

Therefore, we have;

The amount in the account 15 years after when the man deposits another $10,000 = 2 × A

Therefore the initial amount at the 15th year = 2·A + 10000

The

We have;

2·A = A·(1 + r)¹⁵

(1 + r)¹⁵ = 2

1 + r = 2^(1/15)

r = 1 - 2^(1/15) = 0.04729412282

Therefore, we have;

On the 50th year, 50 - 15 = 35 year

$251,894.21 = (2·A + 10000)·(1 + 0.04729412282)³⁵

A = ($251,894.51/((1 + 0.04729412282)³⁵) - 10000)/2 ≈ $19991.1

The initial amount, A ≈ $19991.1.

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