The sum of three no. in g.p is 21 and the sum of their square is 189 find the no.
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Let the G.P. be a , ar , ar² where a is the first term and r is common ratio ,
a + ar + ar² = 21
a ( 1 + r + r² ) = 21 --- eqn. ( 1 )
a² + a²r² + a²r⁴ = 189 ;
a² ( 1 + r² + r⁴ ) = 189 ---eqn. ( 2 )
( 1 + r² + r⁴ ) = ( 1 + r + r² )( 1 - r + r² )
Dividong eqn. ( 2 ) by eqn. eqn. ( 1 )
a( 1 - r + r² ) = 9 ---eqn. ( 3 )
Solving eqn. ( 1 ) and ( 3 ) , we get a = 3 and r = 2
Therefore , the G.P. is 3 , 6 , 12
a + ar + ar² = 21
a ( 1 + r + r² ) = 21 --- eqn. ( 1 )
a² + a²r² + a²r⁴ = 189 ;
a² ( 1 + r² + r⁴ ) = 189 ---eqn. ( 2 )
( 1 + r² + r⁴ ) = ( 1 + r + r² )( 1 - r + r² )
Dividong eqn. ( 2 ) by eqn. eqn. ( 1 )
a( 1 - r + r² ) = 9 ---eqn. ( 3 )
Solving eqn. ( 1 ) and ( 3 ) , we get a = 3 and r = 2
Therefore , the G.P. is 3 , 6 , 12
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