Geometric Progression #2

**Blizzardy** · June 22nd 2011, 08:48 PM

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?

**Sudharaka** · June 22nd 2011, 10:50 PM

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,

$\frac{a(r^{20}-1)}{r-1}=10$

$\frac{a(r^{10}-1)(r^{10}+1)}{r-1}=10$

$S_{10}=\frac{a(r^{10}-1)}{r-1}=\frac{10}{r^{10}+1}$

You have obtained values for $r^{10}$ . Hence there are three possible values that $S_{10}$ could take depending on $r^{10}$ . Hope you can continue.

**Archie Meade** · June 23rd 2011, 04:48 AM

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.

$S_{30}=a+ar+ar^2+ar^3+.......+ar^{29}$

$S_{20}=a+ar+ar^2+ar^3+....+ar^{19}$

$S_{10}=a+ar+ar^2+ar^3+....+ar^9$

We can write 2 equations from the above

$S_{30}-S_{20}=ar^{20}+ar^{21}+....+ar^{29}=r^{20}\left(a+ ar+ar^2+...+ar^9}\right)=r^{20}S_{10}$

Hence

$91-10=81=r^{20}S_{10}$

Also

$S_{20}=S_{10}+ar^{10}+ar^{11}+....+ar^{19}=S_{10}+ r^{10}S_{10}$

$\Rightarrow\ S_{10}\left(1+r^{10}\right)=10$

Therefore, we have the equation

$S_{10}=\frac{10}{1+r^{10}}=\frac{81}{r^{20}}$

This leads to a quadratic equation

$10r^{20}=81\left(1+r^{10}\right)$

$10\left(r^{10}\right)^2-81r^{10}-81=0$

$10x^2-81x-81=0\Rightarrow\ (10x+9)(x-9)=0$

Since $r^{10}$ is an even power and hence positive, the negative solution for x is ruled out.

$r^{10}=9$

Finally

$S_{10}=\frac{81}{\left(r^{10}\right)^2}=1$

**Sudharaka** · June 23rd 2011, 06:50 AM

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, $r^{10}=1\Rightarrow{r=\pm{1}}$

When r=1;

$S_{20}=a+ar+ar^2+ar^3+....+ar^{19}=20a=10 \Rightarrow{a=0.5}$

$S_{30}=a+ar+ar^2+ar^3+.......+ar^{29}=30a=91 \Rightarrow{a=\frac{91}{31}}$

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

When r=-1;

$S_{30}=S_{20}=0$

But we know that, $S_{30}\mbox{ and }S_{20}$ are not equal to zero. Hence we cannot take r=-1 either.

So $r^{10}=1$ cannot be taken.

Case 2: Let, $r^{10}=-\frac{9}{10}$

In this case r would be a complex value and obviously the sums $S_{30}\mbox{ and }S_{20}$ would also be complex values which is again a contradiction.

Therefore we cannot take $r^{10}=-\frac{9}{10}$

The only solution that could be used is, $r^{10}=9$

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