Question 765625: Find the Infinite G.P. whose second term is 2/9 and sum to infinity is unity.
Answer by ramkikk66(644)   (Show Source):
You can put this solution on YOUR website!
For a GP whose 1st term is a and common ratio is r,
Sum to infinity =  ---> Assumption is r < 1
Here, since sum to infinity = 1,
 or 
Second term = a*r = 
Simplifying
Solving using the quadratic solver (see below), we get the 2 solutions as
r = 1/3 or r = 2/3
Correspondingly, a = 1 - r = 2/3 or 1/3
So there are 2 possible GP's:
1) a = 1/3, r = 2/3 i.e. 1/3,2/9,4/27 etc.
2) a = 2/3, r = 1/3 i.e. 2/3,2/9,2/27 etc.
In both cases, the second term is 2/9 and the sum to infinity is 1.
Hope you got it:)
| Solved by pluggable solver: SOLVE quadratic equation with variable |
Quadratic equation  (in our case  ) has the following solutons:
For these solutions to exist, the discriminant  should not be a negative number.
First, we need to compute the discriminant :  .
Discriminant d=9 is greater than zero. That means that there are two solutions:  .
Quadratic expression  can be factored:
Again, the answer is: 0.666666666666667, 0.333333333333333.
Here's your graph:
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