Question

n applicant receives a job offer from two different companies. Offer A is a starting salary of $58,000 and a 3% increase for 5 years. Offer B is a starting salary of $56,000 and an increase of $3,000 per year.

Answer 1

Part A.

The inital salary is $58,000, then we have:

`a_1=58000_{}`

Since we have an increase of 3% each year we know that the second year the salary would be:

`\begin{gathered} a_2=1.03a_1 \\ a_2=1.03\cdot58000 \end{gathered}`

The third year the salary would be:

`\begin{gathered} a_3=1.03a_2 \\ a_3=1.03(1.03)58000 \\ a_3=(1.03)^258000 \end{gathered}`

and so on for year 4 and 5.

Since the increase in salary is only the first five years we conclude that this can't be represented by a geometric series.

For the first five year we can calculate the salary using a geometric sequence with common ratio 1.03, then for the first five years the salary is given by

`a_n=(1.03)^(n-1)_{}\cdot58000\text{ for }1\leq n\leq5`

The salary for the any subsequent year is given by:

`a_n=(1.03)^4\cdot58000\text{ for }n>5`

Part B.

Since we are adding a certain quantity each year we conclude that this offer can be represetend by an algebraic series given by:

`\begin{gathered} b_n=56000+(n-1)3000 \\ b_n=56000+3000n-3000 \\ b_n=3000n+53000 \end{gathered}`

Part C.

After five years the income for offer A is:

`a_5=(1.03)^4\cdot58000=65279.51`

For offer B is:

`b_5=3000(5)+53000=68000_{}`

Therefore after 5 years job offer B has a greater total income.