1. ## Geometric Sequence

In a geometric sequence, the sum of first four terms is 16 times the sum of the following four term, find the common ratio?

Sn =( a(1-r^n)) /1-r

anyone can tell me how to start or resolve this ??

Thank you

2. ## Re: Geometric Sequence

Looks to me to imply:

$\displaystyle S_4=16(S_8-S_4)$. I suspect that this will give you a lot of terms cancelling out, leaving you with a deceptively simple exponential equation to finish (hint: let $\displaystyle x=r^4$). I don't know if there's a better approach.

3. ## Re: Geometric Sequence

Originally Posted by gilagila
In a geometric sequence, the sum of first four terms is 16 times the sum of the following four term, find the common ratio?

Sn =( a(1-r^n)) /1-r

anyone can tell me how to start or resolve this ??

Thank you
I'd use the difference of two squares (repeatedly if necessary) to clear the denominator since there is bound to be a common factor somewhere.

If the sum of the first four terms is larger than the sum of the next four then $\displaystyle |r| < 1$. This will come in useful for checking the answer.

$\displaystyle S_4 = 16(S_8-S_4)$

$\displaystyle S_4 = \dfrac{a(1-r^4)}{1-r} = \dfrac{a(1-r^2)(1+r^2)}{1-r}= a(1+r)(1+r^2)$

$\displaystyle S_8 = \dfrac{a(1-r^8)}{1-r} = \dfrac{a(1-r^4)(1+r^4)}{1-r} = a(1+r^4)(1+r^2)(1+r)$

$\displaystyle S_8 - S_4 = a(1+r^4)(1+r^2)(1+r) - a(1+r)(1+r^2)$

$\displaystyle S_8 - S_4 = \underbrace{a(1+r)(1+r^2)}_{\text{ This is a common factor}}(1+r^4-1) = a(1+r)(1+r^2)r^4$

Now to put this expression back into the original expression:

$\displaystyle \underbrace{a(1+r)(1+r^2)}_{\text{ This is }S_4} = 16\underbrace{a(1+r)(1+r^2)r^4}_{\text{This is }S_8-S_4}$

Do some cancelling and your equation will become clear. Since you have an even power of r there are two possible answers

4. ## Re: Geometric Sequence

Originally Posted by e^(i*pi)
I'd use the difference of two squares (repeatedly if necessary) to clear the denominator since there is bound to be a common factor somewhere.
Would you not save considerable time by stating, after your first line of work, that $\displaystyle 17S_4=16S_8$, then substituting directly into the formula presented in post 1, leading immediately to a factorable quadratic?

5. ## Re: Geometric Sequence

Originally Posted by gilagila
In a geometric sequence, the sum of first four terms is 16 times the sum of the following four term, find the common ratio?

Sn =( a(1-r^n)) /1-r

anyone can tell me how to start or resolve this ??

Thank you
The geometric sequence is

$\displaystyle a,\;\;ar,\;\;ar^2,\;\;ar^3,\;\;ar^4,\;\;ar^5,\;\;a r^6,\;\;ar^7$

$\displaystyle T_5+T_6+T_7+T_8=r^4S_4$

$\displaystyle S_4=16r^4S_4\Rightarrow\ r^4=\frac{1}{16}$

gives the 2 non-trivial solutions.

6. ## Re: Geometric Sequence

Hello, gilagila!

In a geometric sequence, the sum of first four terms
is 16 times the sum of the following four term.
Find the common ratio.

We are given that:.$\displaystyle a_1 + a_2 + a_3 + a_4 \;=\;16(a_5+a_6+a_7+a_8)$

Hence: .$\displaystyle a + ar + ar^2 + ar^3 \;=\;16(ar^4 + ar^5 + ar^6 + ar^7)$

Divide by $\displaystyle a\!:\;\;1 + r + r^2 + r^3 \;=\;16(r^4 + r^5 + r^6 + r^7)$

Factor: .$\displaystyle 1 + r + r^2 + r^3 \;=\;16r^4(1+r+r^2+r^3)$

. . $\displaystyle 16r^4(1+r+r^2+r^3) - (1+r+r^2+r^3) \;=\;0$

Factor: .$\displaystyle (1+r+r^2+r^3)(16r^4-1) \;=\;0$

We have two equations to solve:

$\displaystyle r^3 + r^2 + r + 1 \:=\:0 \quad\Rightarrow\quad r^2(r+1) + (r+1) \:=\:0$

. . $\displaystyle (r+1)(r^2+1) \:=\:0 \quad\Rightarrow\quad \boxed{r \:=\:-1}$

$\displaystyle 16r^4-1 \:=\:0 \quad\Rightarrow\quad r^4 \:=\:\tfrac{1}{16} \quad\Rightarrow\quad \boxed{r \:=\:\pm\tfrac{1}{2}}$

7. ## Re: Geometric Sequence

Originally Posted by Soroban
Hello, gilagila!

We are given that:.$\displaystyle a_1 + a_2 + a_3 + a_4 \;=\;16(a_5+a_6+a_7+a_8)$

Hence: .$\displaystyle a + ar + ar^2 + ar^3 \;=\;16(ar^4 + ar^5 + ar^6 + ar^7)$

Divide by $\displaystyle a\!:\;\;1 + r + r^2 + r^3 \;=\;16(r^4 + r^5 + r^6 + r^7)$

Factor: .$\displaystyle 1 + r + r^2 + r^3 \;=\;16r^4(1+r+r^2+r^3)$

. . $\displaystyle 16r^4(1+r+r^2+r^3) - (1+r+r^2+r^3) \;=\;0$

Factor: .$\displaystyle (1+r+r^2+r^3)(16r^4-1) \;=\;0$

We have two equations to solve:

$\displaystyle r^3 + r^2 + r + 1 \:=\:0 \quad\Rightarrow\quad r^2(r+1) + (r+1) \:=\:0$

. . $\displaystyle (r+1)(r^2+1) \:=\:0 \quad\Rightarrow\quad \boxed{r \:=\:-1}$

$\displaystyle 16r^4-1 \:=\:0 \quad\Rightarrow\quad r^4 \:=\:\tfrac{1}{16} \quad\Rightarrow\quad \boxed{r \:=\:\pm\tfrac{1}{2}}$

With all due respect, within a reasonable time period before your reply, there were already 3 valid solutions presented which guided gilagila towards a solution, whilst allowing him to finish the work for himself. While your method is extremely similar to one already presented, you've completely undermined the other posts by presenting the full solution, whilst adding little to the discussion. I'm not ranting other than to say that I find this slightly irritating.

8. ## Re: Geometric Sequence

Originally Posted by Quacky
With all due respect, within a reasonable time period before your reply, there were already 3 valid solutions presented which guided gilagila towards a solution, whilst allowing him to finish the work for himself. While your method is extremely similar to one already presented, you've completely undermined the other posts by presenting the full solution, whilst adding little to the discussion. I'm not ranting other than to say that I find this slightly irritating.
Soroban is well-known for doing this at other sites...looks like he needs hugs

9. ## Re: Geometric Sequence

thanks for all the people helping me, a simple question but I saw various of solutions, open a new world to me.

thank you very much

10. ## Re: Geometric Sequence

The geometric sequence is

$\displaystyle a,\;\;ar,\;\;ar^2,\;\;ar^3,\;\;ar^4,\;\;ar^5,\;\;a r^6,\;\;ar^7$

$\displaystyle T_5+T_6+T_7+T_8=r^4S_4$

$\displaystyle S_4=16r^4S_4\Rightarrow\ r^4=\frac{1}{16}$

gives the 2 non-trivial solutions.

$\displaystyle T_5+T_6+T_7+T_8=r^4S_4$ <--- how u get this ??

11. ## Re: Geometric Sequence

$\displaystyle T_5+T_6+T_7+T_8=ar^4+ar^5+ar^6+ar^7=r^4(a+ar+ar^2+ ar^3)=r^4S_4$

12. ## Re: Geometric Sequence

Originally Posted by Siron
$\displaystyle T_5+T_6+T_7+T_8=ar^4+ar^5+ar^6+ar^7=r^4(a+ar+ar^2+ ar^3)=r^4S_4$
oops....never noticed that if factories that can be like that, thank you very much