I have this formula:
$\displaystyle {c_n}^2 = \frac{1}{2}{c_{n-1}}^2$
How do I convert this to a series with $\displaystyle \sum$
From n = 1 to n = 10
Hello, Malay!
Edit: You're right! . . . *blush*
. . . . I'll correct it now.
I have this formula: $\displaystyle (c_n)^2 \:= \:\frac{1}{2}(c_{n-1})^2$
How do I convert this to a series with $\displaystyle \sum^{10}_{n=1}$
We have: .$\displaystyle c_n\;=\;\frac{1}{\sqrt{2}}c_{n-1}$
We have a geometric series with first term $\displaystyle c_1 = a$ and common ratio $\displaystyle r = \frac{1}{\sqrt{2}}$
The $\displaystyle k^{th}$ term is: .$\displaystyle c_n\:=\:\frac{a}{2^{\frac{n-1}{2}}} $
Then: .$\displaystyle \sum^{10}_{k=1}\frac{a}{2^{\frac{n-1}{2}}} \;=\;a \cdot \frac{1 - \frac{1}{(\sqrt{2})^{10}}} {1 - \frac{1}{\sqrt{2}}} $
$\displaystyle {c_n}^2 = \frac{1}{2}{c_{n-1}}^2$Originally Posted by chancey
I observed that
$\displaystyle \frac{c_n}{c_{n-1}}=\frac{1}{\sqrt{2}}$
Hence the given sequence is a G.P. with common ratio $\displaystyle \frac{1}{\sqrt{2}}$
Keep Smiling
Malay
I need the series to do this:
Example (not the actual formula, changed it so the numbers were easier)
$\displaystyle c_1 = 1$
$\displaystyle c_2 = \frac{1}{2} 1 = 0.5$
$\displaystyle c_3 = \frac{1}{2} 0.5 = 0.25$
$\displaystyle c_4 = \frac{1}{2} 0.25 = 0.125$
$\displaystyle c_n = ?$
Where each number in the series has to be calculated with the previous number. My question is, can that be put into a series so that i can run a formula once to find $\displaystyle n_{10}$ or do I have to do it 10 times
As explained in the previous discussions this sequence is geometric, thus, if,
$\displaystyle c_1=a$
then,
$\displaystyle c_n=(1/\sqrt{2})^{n-1}a$
or,
$\displaystyle c_n=(-1/\sqrt{2})^{n-1}a$
Where $\displaystyle a$ is the initial term.
Note, it is not possible to find $\displaystyle a$, it can be anything number.
Hello, chancey!
You're expected to be familiar with Arithmetic Series and Geometric Series by nowI need the series to do this:
Example (not the actual formula, changed it so the numbers were easier)
$\displaystyle c_1 = 1$
$\displaystyle c_2 = \frac{1}{2}\cdot 1 = \frac{1}{2}$ . Don't use decimals!
$\displaystyle c_3 = \frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}$
$\displaystyle c_4 = \frac{1}{2}\cdot\frac{1}{4} = \frac{1}{8}$
$\displaystyle c_n = \:?$
Where each number in the series has to be calculated with the previous number.
My question is, can that be put into a series so that i can run a formula once to find $\displaystyle n_{10}$
or do I have to do it 10 times
. . and to be able to "eyeball" a sequence and determine its general form.
This sequence is: .$\displaystyle 1,\;\frac{1}{2},\;\frac{1}{4},\;\frac{1}{8},\; \frac{1}{16},\;\hdots$
How long does it take for you to see that the denominators are doubled each time?
With a little thought, we see that the $\displaystyle n^{th}$ denominator is: $\displaystyle 2^{n-1}$
So the general term is: .$\displaystyle c_n\:= \:\frac{1}{2^{n-1}} $
Of course I can see that, that was just an example, im not really that interested in the answer, but the process is what I want to know.Originally Posted by Soroban
So:
$\displaystyle {c_n}^2 = \frac{1}{2} {c_{n-1}}^2 - 2$
$\displaystyle {c_n} = \sqrt{\frac{1}{2} {c_{n-1}}^2 - 2}$
$\displaystyle {c_n} = (\sqrt{\frac{1}{2} a^2 - 2} )^{n-1}$