A geometric series has three terms. The sum of the three terms is 42. The third term is 3.2 times the sum of other two. What are the terms?

Expert Answers
tiburtius eNotes educator| Certified Educator

`a_1+a_2+a_3=42`                                      (1)

`a_3=3.2(a_1+a_2)`                                    (2) 

Now we put `3.2(a_1+a_2)` instead of `a_3` in first equation.



Now we use the fact that this is geometric sequence which means that `a_n=a_1r^(n-1)` hence we have




Now from (2) we have

`a_1r^2=3.2(a_1+a_1r)` now we put `10/(1+r)` instead of `a_1`


`(10r^2-32r-32)/(1+r)=0` This is equal to 0 only if numerator is equal to 0.


When we solve this equation we get two solutions:

`r_1=-4/5` and  `r_2=4`

For `r_1` we have  `a_1=10/(1-4/5)=50`, `a_2=50 cdot(-4/5)=-40`, `a_3=-40cdot(-4/5)=32`

For `r_2` we have 

`a_1=10/(1+4)=2`, `a_2=2 cdot 4=8`, `a_3=8 cdot 4=32`