Arithmetic Series and Geometric Series

In summary, the problem is to find the sum of an increasing sequence made of 4 positive numbers, where the first three numbers form an arithmetic series and the last three form a geometric series. The last number minus the first number is equal to 30. The conversation discusses various equations and conditions to solve the problem, ultimately arriving at a formula for the sum in terms of the quotient of the geometric progression, which must be between 1 and 2 for the sequence to be positive.
  • #1
Biker
416
52
Here is a question that I have a problem with, It doesn't seem to have a solution:

An increasing sequence that is made of 4 positive numbers, The first three of it are arithmetic series. and the last three are geometric series. The last number minus the first number is equal to 30. Find the sum of this series.

So here is my steps:
Here is the arithmetic series: ##a-d , a, a+d##
The geometric one is: ## a , a+d, ar^2##

So first in order for 2nd one to be geometric series: ##a+d## must equal to ## ar##
We conclude from this that:
## d = a(r-1)##

and the question gave us an equation which is

##ar^2 - (a-d) = 30##

By putting the two equation in each other we get:

## a(r^2 + r - 2) = 30##

Now the sum of the whole sequence is just the sum of the arthmetic plus the last number:
## \text{Sum} = 3a + ar^2 ##
## \text{Sum} = a(r^2 + 3)##and everything pretty much stopped here, I need a third equation to complete this and I didn't find a way to substitute the whole thing is another thing that is solvable. So the question is: Is it solvable in the first place?
 
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  • #2
Solve it in parametric form, and use the information that the terms of the sequence are increasing and positive. You get a formula and a range for the sum.
 
  • #3
ehild said:
Solve it in parametric form, and use the information that the terms of the sequence are increasing and positive. You get a formula and a range for the sum.

Could you give me a start? I had to look for the parametric form to understand what it is ( English isn't my main).
I concluded that r must be in this period ] 1 , 2 ]
So the sum has to be between 52.5 and infinity?

I didnt actually use the parametric form, I substituted the smallest value possible.
 
Last edited:
  • #4
I also guess that something is missing from the problem text. Have you translated it? Series means the sum of a sequence, so I do not think it correct to speak about the first three numbers as arithmetic series. But I am not sure, my English is rather poor.
Is not it possible that the problem meant positive integers?
I would denote the numbers by a, b, c, d, and would use the equations 2b=a+c, c2=bd and d=30+a, and the condition 0<a<b<c<d. All terms of the sequence can be written in terms of a, and you can find some condition for a, so as the sequence is increasing.
You kept both a and r in the final answer. Eliminate one.
 
  • #5
ehild said:
I also guess that something is missing from the problem text. Have you translated it? Series means the sum of a sequence, so I do not think it correct to speak about the first three numbers as arithmetic series. But I am not sure, my English is rather poor.
Is not it possible that the problem meant positive integers?
I would denote the numbers by a, b, c, d, and would use the equations 2b=a+c, c2=bd and d=30+a, and the condition 0<a<b<c<d. All terms of the sequence can be written in terms of a, and you can find some condition for a, so as the sequence is increasing.
You kept both a and r in the final answer. Eliminate one.
Series is a synonyms of sequence. I ensured the translation is correct
You can write it as one equation with substituting the 2nd equation above in the the third one
which will give you
## \text{Sum} = \frac{30(a^2 +3)}{r^2 + r - 2} ##

But we can't get to a final answer, However we got the values of the sum where this is true.. So is there is more of this or ? I don't think there is to be honest
 
  • #6
If r is the quotient of the geometric progression, the sum is
Biker said:
Series is a synonyms of sequence.
No, it is the sum of a sequence( or progression) .http://www.purplemath.com/modules/series.htm
What was the the original text? In what language?
Google translate is wrong sometimes.
Biker said:
I ensured the translation is correct
You can write it as one equation with substituting the 2nd equation above in the the third one
which will give you
## \text{Sum} = \frac{30(a^2 +3)}{r^2 + r - 2} ##

But we can't get to a final answer, However we got the values of the sum where this is true.. So is there is more of this or ? I don't think there is to be honest

You meant ## \text{Sum} = \frac{30(r^2 +3)}{r^2 + r - 2} ##
Stating that r>1 is the quotient of the geometric sequence, the formula above is the solution of the problem.
 
  • #7
ehild said:
If r is the quotient of the geometric progression, the sum is

No, it is the sum of a sequence( or progression) .http://www.purplemath.com/modules/series.htm
What was the the original text? In what language?
Google translate is wrong sometimes.

You meant ## \text{Sum} = \frac{30(r^2 +3)}{r^2 + r - 2} ##
Stating that r>1 is the quotient of the geometric sequence, the formula above is the solution of the problem.
Sorry, was in hurry. Yea I meant that.

I don't use google translate. I guess in mathematics, it means the sum because when I looked it up in the dictionary it said it is the synonym of sequence.
Anyway, we are not in an English course XDWe should state that ##1 < r < 2 ##. If not the first element of the sequence would be actually negative if ## r => 2##
 
  • #8
Biker said:
We should state that ##1 < r < 2 ##. If not the first element of the sequence would be actually negative if ## r => 2##
Yes, that is right.
 

Related to Arithmetic Series and Geometric Series

1. What is an Arithmetic Series?

An Arithmetic Series is a sequence of numbers where each term is obtained by adding a fixed number to the previous term. For example, 1, 3, 5, 7, 9 is an arithmetic series with a common difference of 2.

2. How do you find the sum of an Arithmetic Series?

The sum of an arithmetic series can be found by using the formula: Sn = n/2(a1 + an), where Sn is the sum, n is the number of terms, a1 is the first term, and an is the last term.

3. What is a Geometric Series?

A Geometric Series is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number. For example, 1, 2, 4, 8, 16 is a geometric series with a common ratio of 2.

4. How do you find the sum of a Geometric Series?

The sum of a geometric series can be found by using the formula: Sn = a1(1-r^n)/(1-r), where Sn is the sum, a1 is the first term, r is the common ratio, and n is the number of terms.

5. What is the difference between Arithmetic Series and Geometric Series?

The main difference between arithmetic and geometric series is that in an arithmetic series, each term is obtained by adding a fixed number to the previous term, while in a geometric series, each term is obtained by multiplying the previous term by a fixed number. Additionally, the sum of an arithmetic series is a linear function, while the sum of a geometric series is an exponential function.

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