Help with a Geometric progression.

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In summary: Take the logarithm of both sides and solve for n.In summary, the conversation discusses finding the sum to n terms and the sum to infinity in a geometric progression with given values for the first and fourth terms. The common ratio is determined and used to find the equations for the sum to n terms and the sum to infinity. The last part of the question involves finding the least value of n where the difference between the two sums is less than 0.001. The process for solving this is discussed and a suggestion is made to consult the textbook for a more detailed explanation.
  • #1
misogynisticfeminist
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I need a little help with this problem.

In a geometric progession, the first term is 12 and the fourth term is -3/2. Find the sum to n terms and the sum to infinity. Find also, the least value of n for which the magnitude of the difference between the sum to infinity and to n terms are less than 0.001.

I have first expressed the GP as,

[tex] 12, T_2, T_3, -3/2 [/tex]

I see that the ratio between the 4th and 1st terms is [tex] -\frac{1}{8} [/tex] and this is 3 times the common ration r, which is -1/24. To find the sum to n terms, i get,

[tex] S_n =\frac {12 ( - \frac {1}{24} ^n -1 )}{-1/24-1} [/tex]

and the sum to infinity is 11.52. However the sum to infinity is given as 8 in the answer.

To find the last part of the question, i did,

[tex] 11.52- \frac {12 ( - \frac {1}{24} ^n -1 )}{-1/24-1} = 0.001 [/tex] but it didn't work out to get the answer or n=13.

Thanks a lot for your help.
 
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  • #2
misogynisticfeminist said:
I have first expressed the GP as,

[tex] 12, T_2, T_3, -3/2 [/tex]

I see that the ratio between the 4th and 1st terms is [tex] -\frac{1}{8} [/tex] and this is 3 times the common ration r, which is -1/24. To find the sum to n terms, i get,

Are you sure that's correct?
 
  • #3
OHHH ! it should be

[tex] r^3 = -\frac {1}{8} [/tex]. thanks alot. that should settle it.

edit:

I have found the sum to infinity already and got 8. But have difficulty in the last part where they asked me to find the value of n where the difference between [tex] S_n [/tex] and [tex] S_\infty [/tex] is 0.001

can someone help?
 
Last edited:
  • #4
I don't remember series very well, but I'm surethey offer a great explanation in your textbook. I remember ours had 3 pages to this cause alone.

but as far as I can remember, you set [itex]S_n [/itex] to an errorestimation variable [itex] \epsilon[/itex], then set [itex]S_{\infty} -\epsilon < 0.001 [/tex] and I think you try solvin for n or something like that. Someone else probably has a better answer.
 
  • #5
The equation for [tex] S_\infty [/tex] comes from the equation for [tex] S_n [/tex] by taking the limit as n goes to infinity. Take the difference between the equations for [tex] S_\infty [/tex] and [tex] S_n [/tex] and set it equal to 0.001
 

Related to Help with a Geometric progression.

1. What is a geometric progression?

A geometric progression is a sequence of numbers where each term is found by multiplying the previous term by a constant. This constant is known as the common ratio, and it remains the same throughout the progression.

2. How do you find the common ratio in a geometric progression?

The common ratio can be found by dividing any term in the progression by the previous term. This will give you the same result for all terms in the progression, and that value is the common ratio.

3. What is the formula for finding the nth term in a geometric progression?

The formula for finding the nth term in a geometric progression is an = a1rn-1, where an represents the nth term, a1 represents the first term, and r represents the common ratio.

4. How do you determine if a sequence of numbers is a geometric progression?

A sequence of numbers is a geometric progression if each term is found by multiplying the previous term by a constant, known as the common ratio. You can also check by dividing any term by the previous term and seeing if you get the same value for all terms.

5. What is the sum of a geometric progression?

The sum of a finite geometric progression can be found using the formula Sn = a1(1 - rn) / (1 - r), where Sn represents the sum of the first n terms, a1 represents the first term, and r represents the common ratio. If the progression is infinite, the sum can only be found if the absolute value of the common ratio is less than 1.

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