Clarence
  • Clarence
How do I find the value of c?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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Clarence
  • Clarence
\[\sum_{n=2}^{\infty}7(1+c)^{-n}=2\]
imqwerty
  • imqwerty
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 :)
welshfella
  • welshfella
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)

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welshfella
  • welshfella
Oh in the last equation A1 = 7/a^3
Clarence
  • Clarence
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?
Clarence
  • Clarence
Because what I'm getting for a right now is a really weird number, assuming that I'm doing it right of course
Clarence
  • Clarence
\[a=\frac{ 1 }{ 6 }(2+\sqrt[3]{386-6\sqrt{4137}}+\sqrt[3]{386+6\sqrt{4137}})\]
imqwerty
  • imqwerty
no its wrong :) try putting α=2
Clarence
  • Clarence
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?
imqwerty
  • imqwerty
sorry i did some mistake :) ima do it again
Clarence
  • Clarence
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
imqwerty
  • imqwerty
ok the values of x which i got are really vry bad x~ -1.17 x~1.705
Clarence
  • Clarence
x as in c?
imqwerty
  • imqwerty
sry x=α
Clarence
  • Clarence
And then I'll replace it into what I did before rather than 2?
imqwerty
  • imqwerty
yes :)
Clarence
  • Clarence
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?
Clarence
  • Clarence
so c equals*
alekos
  • alekos
I get α= (1+sqrt15)/2
Clarence
  • Clarence
And c to be that minus 1 then?
alekos
  • alekos
Yes that's right. I just checked my α into the original expression and it does in fact equal 2
alekos
  • alekos
So c = (sqrt15-1)/2
Clarence
  • Clarence
What happened to the 1 plus from earlier? Shouldn't it be (1 + sqrt(15))/2 - 1?
alekos
  • alekos
It boils down to a quadratic 2α^2-2α-7=0
alekos
  • alekos
So α actually might have 2 values
alekos
  • alekos
I just picked the positive value
Clarence
  • Clarence
Ahh, I see, that makes sense
alekos
  • alekos
In answer to your question what you wrote down can be simplified to my value for c
Clarence
  • Clarence
Okay then, well thanks for your help (again) on what was quite a confusing question, much appreciated :)
alekos
  • alekos
No problem

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