Mr. Money has money to give away. He is going to give his money to the best students.

Question

That means that the student must do his homework, pay attention in class, be a thinker in class, and know all the answers the teacher ask. He decided to reward them by given them 3 dollars a day. He said that the first day the student keeps up with the class he will give the student 3 dollars. On the second day he will give the student an additional 3 dollars, bringing their total to 6 dollars, on the third day students will have 9 dollars and so on. The students who are selected will have the opportunity to win a free trip for a week to Disneyland with all expenses paid

A. In order to go on the trip the students will need to accumulate 240 dollars. How many days will it take to earn 240 dollars?

 

B. Students realize it will take too long to get to go on the trip, and instead suggest a different plan. They ask Mr. Money to give them one dollar the first day, then two dollars on the second day, and 4 dollars, and so forth. How many days will take to get to 240 dollars. Write a recursive and explicit formula for this scenario.

Answer 1

A.

S_n = (n/2)(2a₁ + (n - 1)d)

240 = (n/2)(2(3) + (n - 1)3)

240 = (n/2)(6 + 3n - 3)

480 = n(3 + 3n)

480 = 3n² + 3n

0 = 3n² + 3n - 480

0 = n² + n - 160

n ≈ 12                    Quadratic Formula

 

Why would this be too many days to wait?

 

B.

S_n = a₁(1 - rⁿ) / (1 - r)

240 = 1(1 - 2ⁿ) / 1 - 2)

240 = (1 - 2ⁿ / -1

-240 = 1 - 2ⁿ
-241 = - 2ⁿ

241 = 2ⁿ

128 < 241 < 256

2⁷ < 241 < 2⁸

 

So they save a little more than 4 days.

 

Answer 2

A.  The teacher is giving the students $3 per day

     so long as they do as he asks.

 

Money = 3d   where d is the number of days

240 = 3d

240/3 = d

80 = d

 

So long as a student does as the teacher asks it will take

80 days to accumulate $240

 

 

B.    Money = the sum from (n=1) to ∞ of 2^(n-1)               

                  ∞

     Money = ∑ 2^n-1

                  n=1

 

n     2^n-1    Total

--------------------

1      1         1

2      2         3

3      4         7

4      8        15

5     16        31

6     32        63

7     64       127

8     128     255

 

In 8 days a student can accumulate $255