Geometric Series and Sequences Questions!!! HELP!!! *Updated*

Question

I really do not understand the processes involved with these types of problems, so if someone could explain what you did, why you did it, and how can apply it to any word problems in this certain concept, that would be great.

 

 

Your little brother is hard up for cash, and he asks you to lend him 10 cents. You get him to pay you 5% per day interest, compounded daily. You forget the deal until exactly one year later, but you use geometric sequences to calculate how deeply in debt he is to you.

 

a) How much does he owe you for a 365 day year?

b) How much would he owe you if it was a leap year?

 

So for these problems, our teacher gave us the formula:

 

a₁ * ((1-rⁿ)/(1-r))

 

which is the sum of the first n terms of a geometric sequence (r=common ration and a₁₌when n equals one (the first term))

 

If you plug this into your calculator (mine is a TI 83 Plus graphing) , you get the same answers for both equations, and my book tells me that you have to do something with the compound interest formula with this too A=P(1+9(r/n))^(nt). How do you get a different answer for both and how do you use that formula?

 

Thanks so much for your help! I may need more help with other questions, but I'll post them as new ones.

Answer 1

First of all, “A=P(1+9(r/n))^(nt)” has an extraneous “9” in it that doesn’t belong there.

The Compound Interest Formula is A=P(1+(r/n))^(nt) where t is the number of time units, n is the number of times IN EACH TIME UNIT that interest is calculated and added to the account, r is the interest rate PER TIME UNIT, P is the original amount called the Principal, and A is the amount in the account.

For your little brother’s loan:

Day Amount he owes you in cents
0     10
1     10 + 10(5%)
         = 10 + 10(5/100) using definition of Percent
         = 10(1) + 10(0.05)
         = 10(1 + 0.05) using the Distributive Property
         = 10(1.05)
2     (10(1.05))(1) + (10(1.05))(0.05)
         = (10(1.05))(1 + 0.05)
         = 10(1.05)(1.05)
         = 10(1.05)^2
3     (10(1.05)^2)(1) + (10(1.05)^2)(0.05)
         = (10(1.05)^2)(1 + 0.05)
         = 10(1.05)^2(1.05)
         = 10(1.05)^3
…
t      10(1.05)^t
…
365  10(1.05)^365 ≈ $5,421,184.16
366  10(1.05)^366 ≈ $5,692,243.37

Or you could have used the Compound Interest Formula:
A = P(1+(r/n))^(nt) where:
P = 10 cents
t = number of days loan has been active
n = 1 = number of times interest is calculated per day
r = 5% per day = 0.05 per day

A = 10(1+0.05/1)^((1)t) = 10(1.05)^t
which is the same as we got above.

I don’t think there is a way to solve this problem with the formula for the sum of a geometric series. [Show this to your teacher.]

Day Amount owed in cents
0     10
1     10(1.05) = 10 + 10(0.05)
2     10(1.05)^2 = 10(1+r)^2 = 10(1+2r+r^2)
3     10(1+r)^3 = 10(1+3r+3r^2+r^3)
…
t      10(1+r)^t = 10(C(t,0)+C(t,1)r+C(t,2)r^2+…+C(t,t)r^t)

where the Binomial Theorem is used with Binomial Coefficients, C(t,a) = t!/(a!(t-a)!). [Pascal’s Triangle can be used for small t; but finding row 365 would be horrendous.]

There does not seem to be a geometric series in these equations. But let’s try the formula S=a1(1-r^n)/(1-r):

A(365) ≈ $5,421,184.16 from above.

Trying S(0.05) = 10(1-0.05^365)/(1-0.05)
                      ≈ $0.11, which is wrong.

Trying S(1.05) = 10((1.05^365-1)/(1.05-1))
                      ≈ $108,423,681.16, which is wrong.

Answer 2

It is simple once you have gotten the formula

 

a = 10 cent

 

r = 5% = 0.05

 

n = 365  and n = 366 for the leap year

 

 

 

(a)  using the for mula a1 * ((1-r^n) / (1-r))

 

10 * ((1-(0.05)^(365)) / (1-(0.05))

 

use your calculator to solve this

 

 

(b) using the for mula a1 * ((1-r^n) / (1-r))

10 * ((1-(0.05)^(366)) / (1-(0.05))

 

Calculate this with your calculator