Question

How do I find the value of c?

Answer 1

\[\sum_{n=2}^{\infty}7(1+c)^{-n}=2\]

Answer 2

let (1+c)=α
then
then we have-\[\sum_{n=2}^{\infty}7α^{-n}=2\]
\[\frac{ 7 }{ α^2 }+\frac{ 7 }{ α^3 }+\frac{7 }{ α^4 }.......+\frac{ 7 }{ α^{\infty}}=2\]
this is an infinite GP whos summation is 2

can u do it frm here :)

Answer 3

The common ratio of this GP is 7/a^3 divided by 7/a^2 which is 1/a
The sum to infinity =. A1 / (1-r)
Where a1 is the first term (7/a^3 and r. , common ratio, = 1/a
So
A1 / (1 - 1/a) = 2
Solve this for a
Then find c from a = (1+ c)

Answer 4

Oh in the last equation A1 = 7/a^3

Answer 5

So what you're saying is to find a from this \[\frac{ \frac{ 7 }{ a^3 } }{ 1-\frac{ 1 }{ a } }\]
And then find c from a=1+c?

Answer 6

Because what I'm getting for a right now is a really weird number, assuming that I'm doing it right of course

Answer 7

\[a=\frac{ 1 }{ 6 }(2+\sqrt[3]{386-6\sqrt{4137}}+\sqrt[3]{386+6\sqrt{4137}})\]

Answer 8

no its wrong :)
try putting α=2

Answer 9

Sorry, internet issues, so \[\frac{ \frac{ 7 }{ 2^{3} } }{ 1-\frac{ 1 }{ 2 }}\] which equals 7/4 and let that equal 1 + c which means that c would equal 3/4?

Answer 10

sorry i did some mistake :)
ima do it again

Answer 11

It's all good :) Just on that note, I attempted this question earlier before and got 5/2 +/- (3*sqrt(7))/2, with the +/- symbol due to getting a quadratic function, with one of the answers in addition and the other subtraction

Answer 12

ok the values of x which i got are really vry bad
x~ -1.17
x~1.705

Answer 13

x as in c?

Answer 14

sry x=α

Answer 15

And then I'll replace it into what I did before rather than 2?

Answer 16

yes :)

Answer 17

Okay so with 1.705 I got 3.415 and with the -1.17 I got -2.3565. a=1+c so a either equals 2.415 or -3.3565?

Answer 18

so c equals*

Answer 19

I get α= (1+sqrt15)/2

Answer 20

And c to be that minus 1 then?

Answer 21

Yes that's right.
I just checked my α into the original expression and it does in fact equal 2

Answer 22

So c = (sqrt15-1)/2

Answer 23

What happened to the 1 plus from earlier? Shouldn't it be (1 + sqrt(15))/2 - 1?

Answer 24

It boils down to a quadratic
2α^2-2α-7=0

Answer 25

So α actually might have 2 values

Answer 26

I just picked the positive value

Answer 27

Ahh, I see, that makes sense

Answer 28

In answer to your question what you wrote down can be simplified to my value for c

Answer 29

Okay then, well thanks for your help (again) on what was quite a confusing question, much appreciated :)

Answer 30

No problem