If the sequence `(a_(n))` is in GP, such that `a_(4)//a_(6)=1//4 and a_(2)+a_(5)=216,` then `a_(1)` is equal to

Options

A. 12 or 108/7
B. 10
C. 7 or 54/7
D. None of these

Answer

A

Explanation

We have, `(a_(4))/(a_(6))=(1)/(4)`
and `a_(2)+a_(5)=216implies(a_(1)r^(3))/(a_(1)r^(5))=(1)/(4)`
and `a_(1)r+a_(1)r^(4)=216`
where r is the common ratio.
`impliesr^(2)=4 and a_(1)(r+r^(4))=216`
`impliesr=+-2`
now, for r =2, we have
`a_(1)(2+16)=216impliesa_(1)=12`
and for r=-2, we have
`a_(1)(-2+16)=216`
`impliesa_(1)=(216)/(14)=(108)/(7)`