Re: Geometric Progression #2

Quote:

Originally Posted by

**Blizzardy** Thanks a lot mrfantastic!!! =)

But I got a similiar question which I can't solve:

The sum of the first 20 terms of a geometric series is 10 and the sum of the first 30 terms is 91, find the sum of the first 10 terms.

I divided S20 by S10 to eliminate 'a' so as to find 'r':

(r^30 -1) / (r^20 -1) = 91/10

Got to here but I am not sure how to continue. If I expand I will end up with this equation:

10r^30 - 91r^20 + 81 = 0

and the graph of this equation looks kind of weird.

But if I express in this form: 10(r^10)^3 - 91(r^10)^2 +81 = 0

I get r^10 = 9, 1, -9/10

Is there a better way to simplify so I can end up with a simple EQN like the previous question here:

http://www.mathhelpforum.com/math-he...tml#post661852?

Dear Blizzardy,

You have obtained values for . Hence there are three possible values that could take depending on . Hope you can continue.

Re: Geometric Progression #2

Quote:

Originally Posted by

**Blizzardy** Thanks a lot mrfantastic!!! =)

But I got a similiar question which I can't solve:

The sum of the first 20 terms of a geometric series is 10 and the sum of the first 30 terms is 91, find the sum of the first 10 terms.

I divided S20 by S10 to eliminate 'a' so as to find 'r':

(r^30 -1) / (r^20 -1) = 91/10

Got to here but I am not sure how to continue. If I expand I will end up with this equation:

10r^30 - 91r^20 + 81 = 0

and the graph of this equation looks kind of weird.

But if I express in this form: 10(r^10)^3 - 91(r^10)^2 +81 = 0

I get r^10 = 9, 1, -9/10

Is there a better way to simplify so I can end up with a simple EQN like the previous question here:

http://www.mathhelpforum.com/math-he...tml#post661852?

I think that the way the problem was presented,

you could try to find S10 from S30 and S20,

since the difference between these sums is 10 terms.

We can write 2 equations from the above

Hence

Also

Therefore, we have the equation

This leads to a quadratic equation

Since is an even power and hence positive, the negative solution for x is ruled out.

Finally

Re: Geometric Progression #2

Quote:

Originally Posted by

**Blizzardy** Thanks a lot mrfantastic!!! =)

But I got a similiar question which I can't solve:

The sum of the first 20 terms of a geometric series is 10 and the sum of the first 30 terms is 91, find the sum of the first 10 terms.

I divided S20 by S10 to eliminate 'a' so as to find 'r':

(r^30 -1) / (r^20 -1) = 91/10

Got to here but I am not sure how to continue. If I expand I will end up with this equation:

10r^30 - 91r^20 + 81 = 0

and the graph of this equation looks kind of weird.

But if I express in this form: 10(r^10)^3 - 91(r^10)^2 +81 = 0

I get r^10 = 9, 1, -9/10

Is there a better way to simplify so I can end up with a simple EQN like the previous question here:

http://www.mathhelpforum.com/math-he...tml#post661852?

After seeing Archie Meade's post I found a mistake in my previous post. You **cannot** use all the three values you have obtained as I have mistakenly stated.

__Case 1:__ Let,

When r=1;

But 'a' must have a unique value, so we cannot take r=1.

When r=-1;

But we know that, are not equal to zero. Hence we cannot take r=-1 either.

So cannot be taken.

__Case 2:__ Let,

In this case r would be a complex value and obviously the sums would also be complex values which is again a contradiction.

Therefore we cannot take

The only solution that could be used is,