# Arithmetic Series and Geometric Series

• B

## Main Question or Discussion Point

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?

## Answers and Replies

ehild
Homework Helper
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.

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:
ehild
Homework Helper
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.

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 cant 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 dont think there is to be honest

ehild
Homework Helper
If r is the quotient of the geometric progression, the sum is
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.

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 cant 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 dont 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.

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 XD

We should state that $1 < r < 2$. If not the first element of the sequence would be actually negative if $r => 2$

ehild
Homework Helper
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.