Nathematics Actvanced Higher School Cert ficate Examinations by Topics compiled by profectmaths.com, au Page 20 $$ 18 \quad 16 \mathrm{c} $$ Kara deposits an amount of $$ \$ 300000 $$ into an account which pays compound interest of $$ 4 \% $$ per annum, added to the account at the end of each year. Immediately after the interest is added, Kara makes a withdrawal for expenses for the coming year. The first withdrawal is $$ \$ P $$. Each subsequent withdrawal is $$ 5 \% $$ greater than the previous one. Let $$ \$ A_{n} $$ be the amount in the account after the $$ n $$th withdrawal. (i) Show that $$ A_{2}=300000(1.04)^{2}-P[(1.04)+(1.05)] $$ (ii) Show that $$ A_{3}=300000(1.04)^{3}-P\left[(1.04)^{2}+(1.04)(1.05)+(1.05)^{2}\right] $$. (i) Show that there will be money in the account when $$ \left(\frac{105}{104}\right)^{n}

Question

Nathematics Actvanced Higher School Cert ficate Examinations by Topics compiled by profectmaths.com, au Page 20 $$ 18 \quad 16 \mathrm{c} $$ Kara deposits an amount of $$ \$ 300000 $$ into an account which pays compound interest of $$ 4 \% $$ per annum, added to the account at the end of each year. Immediately after the interest is added, Kara makes a withdrawal for expenses for the coming year. The first withdrawal is $$ \$ P $$. Each subsequent withdrawal is $$ 5 \% $$ greater than the previous one. Let $$ \$ A_{n} $$ be the amount in the account after the $$ n $$th withdrawal. (i) Show that $$ A_{2}=300000(1.04)^{2}-P[(1.04)+(1.05)] $$ (ii) Show that $$ A_{3}=300000(1.04)^{3}-P\left[(1.04)^{2}+(1.04)(1.05)+(1.05)^{2}\right] $$. (i) Show that there will be money in the account when $$ \left(\frac{105}{104}\right)^{n}<1+\frac{3000}{P} $$.

Mejia Bolton · Accepted Answer

To ensure Kara's account remains positive after $$ n $$ withdrawals, the condition is: $$ \left(\frac{105}{104}\right)^n < 1 + \frac{3,000}{P} $$

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