Question

70. Suppose that you currently have one credit card with a balance of $10,000 at an annual rate of 24.00% interest. You have stopped adding any additional charges to this card and are determined to pay off the balance. You have worked out the formula bbn = b0r n − R(1 + r+r2 +....+ r n−1), where b0 is the initial balance, bn is the balance after you have maden payments, r= 1 + i, wherei is the monthly interest rate, and R is the amount you are planning to pay each month.
b. Explain why we can rewrite the given formula as bn= b0rn − R(1-rn/1-r).????????

Answer 1

Answer:

$1+r+r^{2}+r^{3}+.....+r^{n-1} = \frac{1-r^{n}}{1-r}$

Step-by-step explanation:

Taking the succession:

$1+r+r^{2}+r^{3}+.....+r^{n-1}$

You can multiply and divide by 1-r without chaging the result:

$\frac{1-r}{1-r} (1+r+r^{2}+r^{3}+.....+r^{n-1})$

Distributing the upper part of the fraction you have:

$\frac{1}{1-r} (1-r+r-r^{2}+r^{2}-r^{3}+r^{3}-r^{4}+.....+r^{n-1}-r^{n})$

As can be seen all the intermediate members will be canceled by a same member with opposite sign, only $(1-r^{n})$ will be left so:

$1+r+r^{2}+r^{3}+.....+r^{n-1} = \frac{1-r^{n}}{1-r}$