if x, 2x+2, 3x+3,...... are in G.P. then the 4th term is
Answers
Given :
x , 2 x + 2 , 3 x + 3 .................. are in G.P
The G.P terms are in the form : a , a r , a r²
Notice that : x = a
2 x + 2 = a r
3 x + 3 = a r²
We know that : a × a r² = a² r²
= ( a r )²
This means that 1 st term × 3 rd term = (2 nd term )² [ For a G.P]
==> x ( 3 x + 3 ) = ( 2 x + 2 )²
==> 3 x² + 3 x = 4 x² + 8 x + 4
[ Using ( a + b )² = a² + b² + 2 a b ]
==> - x² - 5 x - 4 = 0
==> x² + 5 x + 4 = 0
==> x² + 4 x + x + 4 = 0
==> x( x + 4 ) + 1 ( x + 4 ) = 0
==> ( x + 4 )( x + 1) = 0
Either x + 4 = 0 ==> x = -4
or , x + 1 = 0 ==> x = - 1
For x = - 4
The no.s are 2 x + 2 ==> 2 (-4) + 2 ==> -6
The next no. is 3 x + 3 ==> 3(-4)+3 ==> -9
The common ratio is -9 / -6 = 3 / 2
So the 4 th term is : 3 / 2 × 3 rd term
==> 3 / 2 × - 9
==> - 27 / 2
==> - 13 . 5
For x = -1
The no.s are 2 x + 2 ==> 2 (-1) + 2 ==> 0
The next no. is 3 x + 3 ==> 3(-4)+3 ==> 0
So the 4 th term is : r × 3 rd term
==> r × 0
==> 0
The 4 th term can either be 0 or it is -13.5
Hope it helps you
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