1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
marshall27 [118]
3 years ago
9

An employee joined a company in 2009 with a starting salary of $50,000. Every year this employee receives a raise of $1000 plus

5% of the salary of the previous year.
a) Set up a recurrence relation for the salary of this employee n years after 2009.
b) What will the salary of this employee be in 2017?
c) Find an explicit formula for the salary of this employee n years after 2009.
Mathematics
1 answer:
stepladder [879]3 years ago
6 0

Answer:

(a) The required recurrence relation for  the salary of the employee of n years after 2009 is a_n=1.05a_{n-1}+1000.

(b)The salary of the employee will be $83421.88 in 2017.

(c) \therefore a_n=70,000 . \ 1.05^n-20,000

Step-by-step explanation:

Summation of a G.P series

\sum_{i=0}^n r^i= \frac{r^{n+1}-1}{r-1}

(a)

Every year the salary is increasing 5% of the salary of the previous year plus $1000.

Let a_n represents the salary of the employee of n years after 2009.

Then a_{n-1} represents the salary of the employee of (n-1) years after 2009.

Then a_n= a_{n-1}+5\%.a_{n-1}+1000

             =a_{n-1}+0.05a_{n-1}+1000

             =(1+0.05)a_{n-1}+1000

            =1.05a_{n-1}+1000

The required recurrence relation for  the salary of the employee of n years after 2009 is a_n=1.05a_{n-1}+1000.

(b)

Given, a_0=\$50,000

a_n=1.05a_{n-1}+1000

Since 2017 is 8 years after 2009.

So, n=8.

∴ a_8

=1.05 a_7+1000

=1.05(1.05a_6+1000)+1000

=1.05^2a_6+1.05\times 1000+1000

=1.05^2(1.05a_5+1000)+1.05\times 1000+1000

=1.05^3a_5+1.05^2\times 1000+1.05\times 1000+1000

=1.05^3(1.05a_4+1000)+1.05^2\times 1000+1.05\times 1000+1000

=1.05^4a_4+1.05^3\times 1000+1.05^2\times 1000+1.05\times 1000+1000

=1.05^4(1.05a_3+1000)+1.05^3\times 1000+1.05^2\times 1000+1.05\times 1000+1000

=1.05^5a_3+1.05^4\times1000+1.05^3\times 1000+1.05^2\times 1000+1.05\times 1000+1000

=1.05^5(1.05a_2+1000)+1.05^4\times1000+1.05^3\times 1000+1.05^2\times 1000+1.05\times 1000+1000

=1.05^6a_2+1.05^51000+1.05^4\times1000+1.05^3\times 1000+1.05^2\times 1000+1.05\times 1000+1000

=1.05^6(1.05a_1+1000)+1.05^51000+1.05^4\times1000+1.05^3\times 1000+1.05^2\times 1000+1.05\times 1000+1000

=1.05^7a_1+1.05^6\times1000+1.05^51000+1.05^4\times1000+1.05^3\times 1000+1.05^2\times 1000+1.05\times 1000+1000

=1.05^7(1.05a_0+1000)+1.05^6\times1000+1.05^51000+1.05^4\times1000+1.05^3\times 1000+1.05^2\times 1000+1.05\times 1000+1000

=1.05^8a_0+1.05^7\times1000+1.05^6\times1000+1.05^51000+1.05^4\times1000+1.05^3\times 1000+1.05^2\times 1000+1.05\times 1000+1000

=1.05^8a_0+(1.05^7+1.05^6+1.05^5+1.05^4+1.05^3+1.05^2+1.05+1)1000

=1.05^8 \times 50,000+\frac{1.05^8-1}{1.05-1}\times 1000

=1.05^8\times 50,000+20,000(1.58^8-1)

=70,000\times 1.05^8-20,000

≈$83421.88

The salary of the employee will be $83421.88 in 2017.

(c)

Given, a_0=\$50,000

a_n=1.05a_{n-1}+1000

We successively apply the recurrence relation

a_n=1.05a_{n-1}+1000

    =1.05^1a_{n-1}+1.05^0.1000

   =1.05^1(1.05a_{n-2}+1000)+1.05^0.1000

   =1.05^2a_{n-2}+1.05^1.1000+1.05^0.1000

   =1.05^2(1.05a_{n-3}+1000)+(1.05^1.1000+1.05^0.1000)

   =1.05^3a_{n-3}+(1.05^2.1000+1.05^1.1000+1.05^0.1000)

                    ...............................

                   .................................

  =1.05^na_{n-n}+\sum_{i=0}^{n-1}1.05^i.1000

 =1.05^na_0+1000\sum_{i=0}^{n-1}1.05^i

 =1.05^n.50,000+1000.\frac{1.05^n-1}{1.05-1}

 =1.05^n.50,000+20,000.(1.05^n-1)

 =(50,000+20,000)1.05^n-20,000

 =70,000 . \ 1.05^n-20,000

\therefore a_n=70,000 . \ 1.05^n-20,000

You might be interested in
Solve for x <br> 5x - 3 = 12
Mamont248 [21]

Answer:

5x - 3 = 12

5x = 15

<u>x = 3</u>

3 0
3 years ago
Read 2 more answers
Mr. Rivera is making a healthy dessert for his nephew who has diabetes. He wants
kvasek [131]

Answer:

Apple crisp has 10 grams of sugar per serving.  

Greek frozen yogurt has 5grams of sugar per serving.

The recipe with the least amount of sugar per serving is the Greek frozen yogurt.

Step-by-step explanation:

Hi, to answer this question we have to divide the amount of grams of sugar by the number of servings for each case.

Apple crisp: 80 grams ÷8 servings =80 /8 = 10 grams of sugar per serving.

Greek frozen yogurt: 60 grams ÷12 servings =60 /12 = 5 grams of sugar per serving.

So, in conclusion:

Apple crisp has 10 grams of sugar per serving.  

Greek frozen yogurt has 5grams of sugar per serving.

The recipe with the least amount of sugar per serving is the Greek frozen yogurt, because 5<10.

7 0
3 years ago
Derek is working out in order to be in shape for the upcoming football season. He does cardio exercises and strength training. E
Serjik [45]

Answer:

what is the question you only gave the problem

Step-by-step explanation:

7 0
3 years ago
The average (arithmetic mean) weight of the students in the French Club is 150 pounds, and the average weight of the students in
soldier1979 [14.2K]
20 members are in the spanish club
6 0
3 years ago
The figures are similar. Find x
OLga [1]

Answer:

do u have an image of the problem?

Step-by-step explanation:

7 0
3 years ago
Other questions:
  • Are the ratios 6:5 and 5:6 equivalent
    12·1 answer
  • How can you distinguish the base of an isosceles triangle from a leg?
    6·2 answers
  • What does isolate mean in math
    13·2 answers
  • Use the coordinates of the labeled point to find the point-slope equation of the line.
    8·2 answers
  • What is the equation of this line? y=1/2x−3 y=−1/2x−3 y=−2x−3 y=2x−3 
    6·2 answers
  • Ariana charges $15 per ticket for a meet-and-greet plus $5 for parking. Due to a flu, she can only have at most 5 people attendi
    9·1 answer
  • 90 POINTSSSSSSS:
    5·2 answers
  • Tina works 15 hours a week (Monday to Friday). Last week she worked 3 ½ hours on Monday, 4 hours on Tuesday, 2 ⅙ hours on Wednes
    14·1 answer
  • Pls helppppp brooooooooooooooooooooooooo
    6·1 answer
  • Danny’s supermarket has 1 litter water bottles normally priced at $1.50 reduced 25%
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!