Question

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

a dog once a day for a rate of $5 per day. 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. Mrs. Maroney
needs Jayden to walk her dog every day for two weeks. Which plan should she choose? Show the work to justify
your answer

Answer 1

Mrs. Maroney should choose plan 1

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

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