Tina inherits $60,000 and invests it in an account earning interest at a rate of 0.5% per month. Each month, immediately after the interest has been paid, Tina withdraws $800. The amount in the account immediately after the nth withdrawal can be determined using the recurrence relation: $$A_n = A_{n-1}(1.005) - 800,$$ where $n = 1, 2, 3, ...$ and $A_0 = 60,000.$ (a) Use the recurrence relation to find the amount of money in the account immediately after the third withdrawal. (b) Calculate the amount of interest earned in the first three months.

SSCE HSC Mathematics Standard Annuities: Tina inherits $60,000 and invest

Tina inherits $60,000 and invests it in an account earning interest at a rate of 0.5% per month - HSC - SSCE Mathematics Standard - Question 34 - 2020 - Paper 1

Question 34

Tina inherits $60,000 and invests it in an account earning interest at a rate of 0.5% per month. Each month, immediately after the interest has been paid, Tina withd... show full transcript

Worked Solution & Example Answer:Tina inherits $60,000 and invests it in an account earning interest at a rate of 0.5% per month - HSC - SSCE Mathematics Standard - Question 34 - 2020 - Paper 1

Step 1

Use the recurrence relation to find the amount of money in the account immediately after the third withdrawal.

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Answer

To calculate the amount in the account after each withdrawal, we can apply the recurrence relation step by step:

First withdrawal:

$A_1 = A_0(1.005) - 800$
$A_1 = 60,000(1.005) - 800$
$A_1 = 60,300 - 800 = 59,500$
Second withdrawal:

$A_2 = A_1(1.005) - 800$
$A_2 = 59,500(1.005) - 800$
$A_2 = 59,797.50 - 800 = 58,997.50$
Third withdrawal:

$A_3 = A_2(1.005) - 800$
$A_3 = 58,997.50(1.005) - 800$
$A_3 = 59,292.4875 - 800 = 58,492.49$

Therefore, the amount in the account immediately after the third withdrawal is $58,492.49.

Step 2

Calculate the amount of interest earned in the first three months.

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Answer

To find the interest earned in the first three months, we first calculate the total withdrawals and then find out how much the balance has been reduced after these withdrawals:

Total withdrawals after three months:

$800 \times 3 = 2400$
Balance after three months:

Initial amount = $60,000 Amount after third withdrawal =$ 58,492.49
Balance is reduced by:

$60,000 - 58,492.49 = 1,507.51$
Interest earned:

Interest is calculated as the difference between the total withdrawals and the reduction in balance:

$2400 - 1,507.51 = 892.49$

Thus, the amount of interest earned in the first three months is $892.49.

Tina inherits $60,000 and invests it in an account earning interest at a rate of 0.5% per month - HSC - SSCE Mathematics Standard - Question 34 - 2020 - Paper 1

Worked Solution & Example Answer:Tina inherits $60,000 and invests it in an account earning interest at a rate of 0.5% per month - HSC - SSCE Mathematics Standard - Question 34 - 2020 - Paper 1

Use the recurrence relation to find the amount of money in the account immediately after the third withdrawal.

Calculate the amount of interest earned in the first three months.

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