Question

Madyson borrows $1,200 from her father to purchase airplane tickets. The first week after borrowing the money, Madyson repays her father $50. Each week after that, she is able to repay her father $2 more than the previous week, except in the last week, when she finishes repaying the loan. How much is Madyson’s final payment?

Answer 1

If you keep subtracting it, adding $2 each time the final payment would be $78

Answer 2

Answer:  $78 is the Madyson's final payment.

Step-by-step explanation:

Since we have given that

Amount from her father to purchase airplane tickets = $1200

On first week , she paid = $50

According to question it is mentioned that  she is able to repay her father $2 more than the previous week, except in the last week.

So, we will apply "Arithmetic Progression"

a = $50

d = $2

So, we know the formula for "Sum of n terms "

$S_n=\frac{n}{2}(2a+(n-1)d),1200\\\\\frac{n}{2}[2\times 50+(n-1)2]$

Now, we will apply "Quadratic formula "

$\quad x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\\\n=\frac{-49+\sqrt{7201}}{2},\:n=\frac{-49-\sqrt{7201}}{2}\\\\n=17.98\text{ other root is negative so, we will ignore it }\\\\n=17$

So, after 7th week its amount will be

$S_7=\frac{17}{2}[2\times 50+(17-1)2]\\\\S_7=\frac{17}{2}[100+16\times 2]\\\\S_7=\frac{17}{2}[132]\\\\S_7=17\times 66=\ 1122$

so, Madyson's final payment will be

$1200-1122=\ 78$

Hence, $78 is the Madyson's final payment.