Question

Here is a variation on a classic riddle: Jayden has a dog-walking business. He has two plans. Plan 1 includes walking

Answer 1

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

$\ 5\cdot 14=\ 70$

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,

$a_1=1\\ \\a_2=2\\ \\a_3=4\\ \\a_n=2a_{n-1}=a_1\cdot 2^{n-1}\\ \\r=2$

Hence,

$a_{14}=a_1\cdot 2^{14-1}=1\cdot 2^{13}=2^{13}=8,192$

Find the sum $S_{14}=a_1+a_2+\dots+a_{14}:$

$S_{14}=\dfrac{a_1(1-r^n)}{(1-r)}\\ \\S_{14}=\dfrac{1\cdot (1-2^{14})}{1-2}=\dfrac{-16,384}{-1}=16,384\text{ cents }=\ 163.84$

Since $70<$163.84, Mrs. Maroney should choose plan 1