Find a number which, when added to each of 2,6,13, gives three numbers in geometric sequence.  

Expert Answers
lemjay eNotes educator| Certified Educator

To start, let's assign a variable to the number. Let it be x.

And if adding x to 2, 6 and 13 results to a geometric series, we would have:

`2+x` , `6+x` , and `13+x`

Note that in geometric series, the ratio between two consecutive terms are the same and it is called as a common ratio (r). So,

`r = (6+x)/(2+x)`               and              `r=(13+x)/(6+x)`

To solve for x, set the two equations equal to each other.



Then, cross multiply.



To combine like terms, move `x^2` from the left to the right side of the equation.



Also, move 12x to the right.



Then, isolate x.




Hence, the number that is added to 2,6 and 13, which gives three numbers in geometric series, is `10/3` .