Hi

"The sum of an infinite geometric series is twice its first term.

 Find the common ratio of the series

 =  when |r| < 1

a/(1-r) = 2a

 1/(1-r) = 2

   1 = 2 - 2r

   2r = 1

    r = 1/2



In general: the sum of a geometric series is:

  

As the 'infinite' sum being just twice it's first term, was safe, in my mind,

to assume that r was a fraction and r^n would become so...insignificant that (1-r^n)= 1  

For ex:  (1/2)^20 = .0000001 demonstrates that processs

   

Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
 the formula for the sum of the first n terms of a geometric series is equal to:





S[n] = (a[1] * (1-r^n)) / (1-r) where a[1] is the first term in the series.





an example:





n = 3 and r = .7 and a[1] = 500





S[3] = (500 * (1-.7^3)) / (1-.7)





this becomes equal to 1095.





We can verify this by adding the elements together.





the nth element in a geometric series is given by the equation a[n] = a[1]*r^(n-1).





a[1] = 500 * .7^0 = 500

a[2] = 500 * .7^1 = 350

a[3] = 500 * .7^2 = 245





sum is 500 + 350 + 245 = 1095





as n approaches infinity, the formula of:





S[n] = (a[1] * (1-r^n)) / (1-r) where a[1] is the first term in the series.





becomes:





S[infinity] = a[1] / (1-r) where a[1] is the first term in the series.





r^n approaches 0 as n approaches infinity because r has to be smaller than 1.





this makes the expression (1-r^n) approach 1 as n approaches infinity.





an example:





let r = .9





.9^10 = .35

.9^100 = .000027

.9^1000 = 1.75 * 10^-46

.9^10000 = 0 on my calculator.   it's not really 0 but it's so small that the calculator is not able to show it so it rounds the answer to 0.





at any rate, the formula for the sum of a geometric series as n approaches infinity is equal to:





S[infinity] = a[1] / (1-r) where a[1] is the first term in the series.





your problem states:



-----

The sum of an infinite geometric series is twice its first term. Find the common ratio of the series.

-----





if the sum of the infinite geometric series is twice its first term, then we get:





S[infinity] = 2*a[1]





we can replace S[infinity] in the equation of:





S[infinity] = a[1] / (1-r) where a[1] is the first term in the series.





by 2*a[1] to get:





2 * a[1] = a[1] / (1-r)





if we multiply both sides of this equation by (1-r) and we divide both sides of this equation by 2 * a[1] then we get:





(1-r) = a[1] / (2 * a[1])





we simplify this to get:





(1-r) = 1/2





we add r to both sides of this equation and we subtract 1/2 from both sides of this equation to get:





r = 1/2





let's see if this can be confirmed.





we assume r = 1/2 as we just calculated.





we let a[1] be any number chosen at random.





let's try 1469





S[infinity] = a[1] / (1-r) = 1469 / (1-(1/2) = 1469 / (1/2) = 2938





2938 is twice as large as 1469 so the equation is good.





it will work in all equations, since the general form would be:





S[infinity] = a[1] / (1 - (1/2) which always winds up being:





S[infinity] = a[1] / (1/2).





reference for formulas used is in the following link:





http://people.richland.edu/james/lecture/m116/sequences/geometric.html





Remember that r has to be smaller than 1 if we want to get the sum of the geometric series as n approaches infinity.

























 

RELATED QUESTIONS

The sum of infinite geometric series with first term 3 and second term 0.75... (answered by htmentor)

I hope you can help sove this question;

The first term in an arithmetic series is 16... (answered by vleith)

find the common ratio of an infinite geometric series with first term 6 and with sum... (answered by funmath)

The sum of an infinite geometric series is 5/2 and the first term is 3. Find the common... (answered by stanbon)

The sum of an infinite geometric series is 108, while the sum of the first 3 terms is... (answered by greenestamps)

an infinite geometric series with second term -8/9 and sum 2. what is the first... (answered by ikleyn)

An infinite geometric series has the first term 195 and a common ratio -0.59. 
What is... (answered by KMST)

An infinite geometric series has first term 70 and common ratio -0.23.

To 2 decimal... (answered by Fombitz)

An infinite geometric series has first term 70 and common ratio -0.23.

To 2 decimal... (answered by KMST)