Question

You have found two job offers for the beginning of the summer. Each job is only 10 days long. Job Offer #1: The offer is to receive $0.25 for accepting the job. After day one you have made a total of $1, $4 total after two days of work, $16 total after three days, etc. Job Offer #2: The offer is $16 per day with no acceptance bonus. Write an equation to represent Job Offer #1 and Write an equation to represent Job Offer #2

Answer 1

`f(n) = (4^n-0.25)/(3)` --- Job offer 1

`g(n) = 16n` --- Job offer 2

`1 \le n \le 10`

Explanation:

Given

Job offer 1:

Offer = $0.25

`Day 1 = \$1\ Day 2 = \$4\ Day 3 = \$16......`

Job offer 2:

Daily = $16

Required

Determine the equation for both jobs

For job offer 1:

Considering the pay for day 1, day 2,....

The sequence shows a geometry progression where the payment between subsequent days is a product of 4 by the payment of the previous day.

The sequence can be represented as:

`T_1 = 1; T_2 = 4; T_3 = 16`

The common ratio (r) is:

`r = (T_2)/(T_1) = (4)/(1) = 4`

The sum of n terms is the total salary received in n days.

`S_n = (a(r^n - 1))/(r-1)`

`S_n = (1 * (4^n - 1))/(4-1)`

`S_n = (4^n - 1)/(3)`

So, the equation for job offer 1 is:

`f(n) = Offer + S_n`

`f(n) = 0.25+ (4^n - 1)/(3)`

Take LCM

`f(n) = (0.75 + 4^n - 1)/(3)`

Collect like terms

`f(n) = (4^n - 1+0.75)/(3)`

`f(n) = (4^n-0.25)/(3)`

For job offer 2:

Daily payment of $16 implies that job offer 2 pays 16n for n days.

So:

`g(n) = 16n`

In both cases:
`1 \le n \le 10`