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 "
[tex]S_n=\frac{n}{2}(2a+(n-1)d),1200\\\\\frac{n}{2}[2\times 50+(n-1)2]<1200\\\\\frac{n}{2}[100+2n-2]<1200\\\\\frac{n}{2}[98+2n],1200\\\\n^2+49n<1200[/tex]
Now, we will apply "Quadratic formula "
[tex]\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[/tex]
So, after 7th week its amount will be
[tex]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[/tex]
so, Madyson's final payment will be
[tex]1200-1122=\$78[/tex]
Hence, $78 is the Madyson's final payment.
The final amount that will be paid by Madison if she borrows $1,200 from her dad will be $78.
Given to us
Madyson borrows $1,200 from her father to purchase airplane tickets.
Madyson repays her father $50 in the first week,
Madyson repay her father $2 more than the previous week, except in the last week,
We know that Madison has taken $1,200 from her father and she paid $50 in the first week while she is paying $2 more than the previous week.
Let's assume that Madison pays her father n number of times therefore, we can assume the process to be an arithmetic sequence.
The difference of the arithmetic series is 2 as she is paying $2 more than the previous week each time,
[tex]S_n = \dfrac{n}{2}(2a+(n-1)d)[/tex]
[tex]1200\leq \dfrac{n}{2}(2(50)+(n-1)2)\\\\2400\leq {n}(100+2n-2)\\\\2400\leq 100n +2n^2 -2n\\\\2400\leq 98n +2n^2\\\\0 \leq 2n^2 +98n -2400\\\\0\leq n^2 + 49n -1200[/tex]
We got the quadratic equation finding the roots we will get,n = 17.92935 and n = -66.92935
As the second root is negative, therefore, taking the first root,
Thus, it will take 17 weeks before her final payment.
The amount paid by Madison for 17 weeks
Using the arithmetic sequence,
[tex]S_n = \dfrac{n}{2}(2a+(n-1)d)[/tex]
Substitute the values,
[tex]S_n = \dfrac{17}{2}[2(50)+(17-1)2][/tex]
[tex]S_n = \$1,122[/tex]
Thus, Madison will pay $1,122 of the total amount till her 17th week.
The Madison final amount
The final amount Madison will pay in the 18th week will be,
$1,200 - $1,122 = $78
Hence, the final amount that will be paid by Madison in the 18th week if she takes $1,200 from her dad will be $78.
Learn more about Arithmetic Sequence:
https://brainly.com/question/16130064
