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March 28th, 2015, 01:58 AM   #1
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Geometric progression

Hello

I've got a problem with solving this geometric progression example:

-2 +8 -32+ ..... +x =26214 <----- We are looking for value of x

I'm writing a math essay on Monday and I need to learn this example.
So, I'd like to ask for some help.

Thanks in advance.

Last edited by skipjack; March 28th, 2015 at 08:19 PM.
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March 28th, 2015, 02:17 AM   #2
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You have all the neccessary information. ($\displaystyle b_1=-2, \; \; q=-4, \; \; S_n = 26214$ and you need to find $\displaystyle b_n$).

Use Sum formula for geometric progression, $\displaystyle S_n = \frac{b_1 (q^n -1)}{q-1}$ to find n, then find x, $\displaystyle b_n=b_1 q^{n-1}$
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March 28th, 2015, 02:31 AM   #3
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Quote:
Originally Posted by tahirimanov19 View Post
You have all the neccessary information. ($\displaystyle b_1=-2, \; \; q=-4, \; \; S_n = 26214$ and you need to find $\displaystyle b_n$).

Use Sum formula for geometric progression, $\displaystyle S_n = \frac{b_1 (q^n -1)}{q-1}$ to find n, then find x, $\displaystyle b_n=b_1 q^{n-1}$
Thanks for reply

I'm having problem with dragging out n from Sum Formula $\displaystyle S_n = \frac{b_1 (q^n -1)}{q-1}$ ,can you or someone else please show/explain how to drag out n= please ?
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March 28th, 2015, 02:54 AM   #4
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Quote:
Originally Posted by atest0504 View Post
Thanks for reply

I'm having problem with dragging out n from Sum Formula $\displaystyle S_n = \frac{b_1 (q^n -1)}{q-1}$ ,can you or someone else please show/explain how to drag out n= please ?
$\displaystyle 26214 = \frac{-2 \times ((-4)^n -1)}{-5}$

$\displaystyle 65536=(-4)^n \Rightarrow n=8$
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March 28th, 2015, 03:25 AM   #5
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Quote:
Originally Posted by tahirimanov19 View Post
$\displaystyle 26214 = \frac{-2 \times ((-4)^n -1)}{-5}$

$\displaystyle 65536=(-4)^n \Rightarrow n=8$
How did you get from 26214 to 65536 (2 to the 16th power) ?
Where did go -2 -1 and -5 from this formula: $\displaystyle 26214 = \frac{-2 \times ((-4)^n -1)}{-5}$
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March 28th, 2015, 05:38 AM   #6
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$\displaystyle 26214 = \frac{-2 \times ((-4)^n -1)}{-5}$

$\displaystyle 26214 \times 5/2 = \frac{-1 \times ((-4)^n -1)}{-1} = (-4)^n - 1$

$\displaystyle 131070/2 + 1 = (-4)^n$

$\displaystyle 65535 + 1 = (-4)^n$

$\displaystyle 65536 = (-4)^n$

$\displaystyle 4^8 = (-4)^n = 4^n\ $ (if $n$ is even)

$\displaystyle n = 8$
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March 28th, 2015, 06:12 AM   #7
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Is the final result $\displaystyle X=-2 (-4)^{8-1} = 32768$ ?
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March 28th, 2015, 09:57 AM   #8
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Is the final result $\displaystyle X=-2 (-4)^{8-1} = 32768$ ?
Can someone please confirm if I calculated it right asap and if I failed please give me correct result.I'm in kind of hurry.
Thanks..
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March 28th, 2015, 11:17 AM   #9
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Quote:
Originally Posted by atest0504 View Post
Is the final result $\displaystyle X=-2 (-4)^{8-1} = 32768$ ?

x = 32,768 is the answer.$\displaystyle \ \ \ \ \ \ \ $Keep it lower case x.



By the way, -2 + 8 - 32 + ... + x is a geometric series.

And -2, 8, -32, ... , x is the corresponding geometric progression.
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March 28th, 2015, 04:29 PM   #10
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$\displaystyle a=-2$

$\displaystyle r=\frac{8}{-2}=-4$

$\displaystyle S_n=\frac{a(r^n-1)}{r-1}$

$\displaystyle 26214=\frac{-2(-4^n-1)}{-4-1}$

$\displaystyle -131070=-2(-4^n-1)$

$\displaystyle 65535=-4^n-1$

$\displaystyle (-4)^n=65536$

$\displaystyle (-4)^n=4^8$

$\displaystyle n=8$

$\displaystyle T_n=ar^{n-1}$

$\displaystyle T_8=ar^7$

$\displaystyle =-2(-4)^7$

$\displaystyle =32768$
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Last edited by jiasyuen; March 28th, 2015 at 04:34 PM.
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