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BITSAT2016MathematicsSequences and SeriesActual

If a, b , and c are in A.P. and also a - 2 b, 2 b- 3 c, 3 c- a are in A.P., then

Options

  1. Aa , b , c , are in H.P.
  2. Ba, 2 b, 3 c are in A.P.
  3. Ca , b , c are the sides of a triangle
  4. Dnone of the above

Correct answer

C. a , b , c are the sides of a triangle

Step-by-step solution

( a , b , c ) are in A.P. ( aligned & 2 b = a + c & b ²= ( ac ) & b ²= ac a , b , c are in G.P. & a - 2 ~b , 2 ~b - 3 c , 3 c - a are in A.P. aligned ) ( 2( 2 b- 3 c)=( a- 2 b)+( 3 c- a) ) ( 3 2 ~b =3 3 c 2 ~b =3 c ) Now, (b²=a c b²=a 2 b 3 b= 2 a 3 , c= 4 a 9 ) ( aligned & i.e., a=a, b= 2 a 3 , c= 4 a 9 & a: b: c=1: 2 3 : 4 9 =9: 6: 4 aligned ) Since, sum of any two is greater than the (3^ rd , a , b , c ) form a triangle.

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