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The next term of the G.P. x ,x^2+2,and x...

The next term of the G.P. `x ,x^2+2,and x^3+10` is

A

0

B

6

C

`(729)/(16)`

D

54

Text Solution

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The correct Answer is:
C, D
According to the question, `x,x^(2)+2" and "x^(3)+10` are in GP.
So, `(x^(2)+2)^(2)=x(x^(3)+10)=0`
`implies x^(4)+4+4x^(2)-x^(4)-10x=0`
`implies 4x^(2)-10x+4=0`
`implies 2x^(2)-5x+2=0`
`implies 2x^(2)-4x-x+2=0`
`implies 2x(x-2)-1(x-2)=0`
`implies(x-2)(2x-1)=0`
`implies x=2` or `x=(1)/(2)`
For`x=2`,first 3 terms are2,6,18.
So, 4th term of GP `=2*(3)^(3)=54`
For `x=(1)/(2)`, first 3 terms are `(1)/(2), (9)/(4),(81)/(8)`.
So, `T_(4)=(1)/(2)((9)(2))^(3)=(1)/(2)xx(729)/(16)`
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