Sum of this geometric sequence doesn't make sense

In summary, a geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a constant number, known as the common ratio. The sum of a geometric sequence refers to the total of all the terms in the sequence and can be calculated using a specific formula. The sum may not make sense if the common ratio is equal to 1, greater than 1, or if the sequence does not follow the formula for a geometric sequence. To determine if a sequence is geometric, you can check if each term is obtained by multiplying the previous term by a constant number or by calculating the common ratio. Geometric sequences are significant in science as they can be used to model natural phenomena and have applications in various fields.
  • #1
It_Angel
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?
 
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  • #2
It_Angel said:
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]

It_Angel said:
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
Yeah you got the typo.

Why is a not 2, as per tn=a*r^n?
 
  • #4
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
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
It_Angel said:

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
 

What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a constant number, known as the common ratio. The general form of a geometric sequence is a, ar, ar2, ar3, ... where a is the first term and r is the common ratio.

What is the sum of a geometric sequence?

The sum of a geometric sequence refers to the total of all the terms in the sequence. This can be calculated using the formula S = a(1-r^n)/1-r, where S is the sum, a is the first term, r is the common ratio, and n is the number of terms in the sequence.

Why doesn't the sum of this geometric sequence make sense?

There could be several reasons why the sum of a geometric sequence doesn't make sense. One reason could be that the common ratio is equal to 1, which would result in an infinite sum. Another reason could be that the common ratio is greater than 1, which would also result in an infinite sum as the terms would continue to get larger and larger. Additionally, if the sequence does not follow the formula for a geometric sequence, the sum may not make sense.

How can I tell if a sequence is geometric?

To determine if a sequence is geometric, you can check if each term is obtained by multiplying the previous term by a constant number. You can also calculate the common ratio by dividing any term by the previous term. If the common ratio is the same for each term, then the sequence is geometric.

What is the significance of geometric sequences in science?

Geometric sequences are important in science as they can be used to model natural phenomena such as population growth, radioactive decay, and cell division. They also have applications in finance, physics, and computer science. Understanding geometric sequences can help scientists make predictions and solve problems in various fields.

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