A geometric sequence has 3 first terms a,b,b2. If a and b are roots of the quadratic equation 2x2+kx+6=0, then the fourth term of the sequence and the value of k respectively are....
Explanation
A geometric sequence has a ratio between its terms, or the value of the ratio between terms
ab=bb2
ab=1b
a=1
The quadratic equation 2x2+kx+6=0 has roots x1=a=1 and x2=b
Product of roots
x1⋅x2=26
(1)x2=3
b=3
Sum of roots
x1+x2=2−k
1+3=2−k
4=2−k
k=−8
Therefore the geometric sequence is
a,b,b2=1,3,9,...
Then the 4th term is
u4=ar3
u4=(1)(3)3
u4=27
Thus the values of u4 and k respectively are 27 and −8.
