To prove: a, (a – b) and (d – c) are in GP.Given: a, b, c are in AP, and a, b, d are in GPProof: As a,b,d are in GP thenb2 = ad … (i)As a, b, c are in AP2b = (a + c) … (ii)Considering a, (a – b) and (d – c)(a – b)2 = a2 – 2ab + b2= a2 – (2b)a + b2From eqn. (i) and (ii)= a2 – (a+c)a + ad= a2 – a2 - ac + ad= ad – ac(a – b)2 = a (d – c)From the above equation we can say that a, (a – b) and (d – c) are in GP.

Q. 195.0( 2 Votes )

If a, b, c are in

Class 11th RS Aggarwal - Mathematics 12. Geometrical Progression

Answer :

To prove: a, (a – b) and (d – c) are in GP.

Given: a, b, c are in AP, and a, b, d are in GP

Proof: As a,b,d are in GP then

b² = ad … (i)

As a, b, c are in AP

2b = (a + c) … (ii)

Considering a, (a – b) and (d – c)

(a – b)² = a² – 2ab + b²

= a² – (2b)a + b²

From eqn. (i) and (ii)

= a² – (a+c)a + ad

= a² – a² - ac + ad

= ad – ac

(a – b)² = a (d – c)

From the above equation we can say that a, (a – b) and (d – c) are in GP.

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