Hi! We started a sequence chapter in my math class today, and were given a worksheet to complete. I understood most of it, but got stumped on these problems. Can someone please explain?

Question

1. Suppose that you are responsible for predicting the cost of constructing a new multi-story office building. The cost per square meter of floor space for constructing the higher stories increases because it is more difficult to build them. You find out that the first story costs $400 per square meter and the second story cost $425. Assume that costs per square meter form an arithmetic sequence.

a. What is the cost per square meter for the third, fourth stories?

b. The building is 48 stories high. How much per square meter will the top story cost?

c. What is the total cost per square meters for the building all 48 stories?

2. Suppose there is a cake that you are sharing with your sibling. On the first day, you each eat 1/3 of the cake. That means there is 1/3 left over. Then next day you each eat 1/3 of what is left.

a. How much do you eat on the second day?

b. If you follow this pattern for a total of 5 days, how much do you eat on the fifth day?

c. If you keep following this pattern do you ever run out of cake?

d. If you follow this pattern forever, how much cake have you eaten?

Thank you! :)

Answer 1

1)

a_n = 400 + (n - 1)25, use this for a. and b.

S₄₈ = (48/2)(2(400) + 48 - 1)(25)

2)

a_n = (1/3)^n-1, use this for a and b

c and d try large values of n and discover.

Answer 2

Question 1:

It is arithmetic sequence question.

As A₁=400, D=25 and we know A_n=A₁+(n-1)D

So, Q1-a: A₃ =A₁+2D=400+50=450

A₄=A₁+3D=400+75=475

Q1-b: A₄₈=A₁+47D=400+47x25=1575

Q1-c: SUM₄₈=A₁+A₂+A₃+.......+A₄₈=48/2(2x400+47x25)=24x(800+1175)=47400

Question 2:

This is Geometric sequence question. Its A₁=1/3 and R=1/3

Q2-a: As An=A₁R^n-1, A₂=1/3 x (1/3)^2-1=1/9

Q2-b: A₅=1/3x(1/3)^5-1=1/3x(1/3)⁴=1/243

Q2-c: As An=A₁R^n-1, A₁ and R none equal to 0, An never will be 0. (The cake cannot run out,)

Q2-d: Due to

a=A₁=1/3 and r=R=1/3

if n go to infinite (1/3)ⁿ will go to 0. the sum will be 1/2

So, the answer for this question is 1/2.