• Support PF! Buy your school textbooks, materials and every day products Here!

Sum of this geometric sequence doesn't make sense!

  • Thread starter It_Angel
  • Start date
  • #1
6
0

Homework Statement



14
Ʃ 2(4/3)^n
n=1

Homework Equations



Sn=a(1-r^n)/(1-r)



The Attempt at a Solution



2(1-[4^14]/[3^14])/(-1/3)=330.74

However, the answer sheet gives ~441 as the answer, and I confirmed it by doing it by hand. Why is the equation not working? What's wrong?
 
Last edited:

Answers and Replies

  • #2
46
0
14
Ʃ 2(4/3)^2
n=1
I'm assuming that second 2 is a typo and should be an n.
[itex]\sum ^{14}_{n=1} 2(\frac{4}{3})^n[/itex]

However, the answer sheet gives ~441 as the answer, and I confirmed it by doing it by hand. Why is the equation not working? What's wrong?
I believe the equation is working. [itex]a[/itex] represents the first term in the series. In this case, what is [itex]a[/itex]?
 
  • #3
6
0
Yeah you got the typo.

Why is a not 2, as per tn=a*r^n?
 
  • #4
46
0
The sum of a geometric series is defined as:
[itex]a+ar+ar^2+ar^3+...+ar^{n-1} = a\frac{1-r^n}{1-r}[/itex]

If n started at 0, then a would be 2.
Since n starts at 1, in order to form a geometric series we must group it as following:
[itex]\frac{8}{3} + \frac{8}{3}(\frac{4}{3}) + \frac{8}{3}(\frac{4}{3})^2 + ... + \frac{8}{3}(\frac{4}{3})^{13}[/itex]
 
  • #5
Mentallic
Homework Helper
3,798
94
It_Angel, you might not have put this together for yourself so I'll just mention it.

The reason the geometric sum

[tex]a+ar+ar^2+...+ar^n = a\frac{1-r^{n+1}}{1-r}[/tex]

Is because we can simply factor out an "a" on the left side, and then if we compare both sides,

[tex]a(1+r+r^2+...+r^n)=a\left(\frac{1-r^{n+1}}{1-r}\right)[/tex]

Clearly we can just divide both sides by "a" to get what the geometric sum (starting from 1) is equal to.

Anyway, the moral of the story is if you can't figure out what a should be, all you need to do is factor out some value such that the geometric sum inside the factor begins at 1, and then you know the value you factored out must be a. Or even more easily: Whatever the first value of the sum is, that is equal to a.
 
  • #6
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728

Homework Statement



14
Ʃ 2(4/3)^n
n=1

Homework Equations



Sn=a(1-r^n)/(1-r)



The Attempt at a Solution



2(1-[4^14]/[3^14])/(-1/3)=330.74

However, the answer sheet gives ~441 as the answer, and I confirmed it by doing it by hand. Why is the equation not working? What's wrong?
Two problems: (i) incorrect evaluation of result; and (ii) incorrect formula. We have
[tex] a \sum_{n=0}^N r^n = a \frac{1-r^{N+1}}{1-r},\\
a \sum_{n=1}^N r^n = a \frac{r - r^{N+1}}{1-r}.[/tex]
The formula starting at n = 1 is a bit different from that starting at n = 0.

Anyway, I get ##2 \sum_{n=0}^{14} (4/3)^n \doteq 442.9854833,## while ##2 \sum_{n=1}^{14} (4/3)^n \doteq 440.9854833.## I cannot get your 330.74 from either formula.

RGV
 

Related Threads on Sum of this geometric sequence doesn't make sense!

  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
4
Views
2K
Replies
8
Views
1K
Replies
5
Views
1K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
6
Views
4K
  • Last Post
Replies
8
Views
2K
Top