Answer:
Mrs. Maroney should choose plan 1
Step-by-step explanation:
Plan 1:
Plan 1 includes walking a dog once a day for a rate of $5 per day.
In 2 weeks (14 days), Jayden will earn
[tex]\$5\cdot 14=\$70[/tex]
Plan 2:
Plan 2 also includes one walk a day but charges 1 cent for 1 day, 2 cents for 2 days, 4 cents for 3 days, and 8 cents for 4 days and continues to double for each additional day.
So,
[tex]a_1=1\\ \\a_2=2\\ \\a_3=4\\ \\a_n=2a_{n-1}=a_1\cdot 2^{n-1}\\ \\r=2[/tex]
Hence,
[tex]a_{14}=a_1\cdot 2^{14-1}=1\cdot 2^{13}=2^{13}=8,192[/tex]
Find the sum [tex]S_{14}=a_1+a_2+\dots+a_{14}:[/tex]
[tex]S_{14}=\dfrac{a_1(1-r^n)}{(1-r)}\\ \\S_{14}=\dfrac{1\cdot (1-2^{14})}{1-2}=\dfrac{-16,384}{-1}=16,384\text{ cents }=\$163.84[/tex]
Since $70<$163.84, Mrs. Maroney should choose plan 1
