Find all possible calues of the common ratio of the geometric sequence a,b,6, ... if a+b = 1.
The general form of a geometric sequence is:
a , a r , a r^2
In your case :
a + b = 1
b = 1 - a
b = a r alsoo
1 - a = a r
1 = a r + a
1 = a ( r + 1 ) Divide both sides with ( r + 1 )
1 / ( r + 1 ) = a
a = 1 / ( r + 1 )
6 = a r ^ 2
6 = [ 1 / ( r + 1 ) ] * r ^ 2
r ^ 2 / ( r + 1 ) = 6 Multiply both sides with ( r + 1)
r ^ 2 = 6 ( r + 1 )
r ^ 2 = 6 r + 6
r ^ 2 - 6 r - 6 = 0
Solutions of this equation are:
r = 3 - sqrt ( 15 ) =
-0,8729833462074168851792653997824
and
r = 3 + sqrt ( 15 ) = -0,8729833462074168851792653997824
a = 1 / ( r + 1 )
In tis case :
a = 1 / ( 3 - sqrt ( 15 ) + 1 )
a = 1 / ( 4 - sqrt ( 15 ) )
and
a = 1 / ( 3 + sqrt ( 15 ) + 1 )
a = 1 / ( 4 + sqrt ( 15 ) )