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,

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

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

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

You have obtained values for $\displaystyle r^{10}$. Hence there are three possible values that $\displaystyle S_{10}$ could take depending on $\displaystyle r^{10}$. 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.

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

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

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

We can write 2 equations from the above

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

Hence

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

Also

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

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

Therefore, we have the equation

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

This leads to a quadratic equation

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

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

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

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

$\displaystyle r^{10}=9$

Finally

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

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

When r=1;

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

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

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

When r=-1;

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

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

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

__Case 2:__ Let, $\displaystyle r^{10}=-\frac{9}{10}$

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

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

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