Question

three positive numbers forms and increasing GP if the middle term of the GP is doubled then New numbers are in AP then common ratio of GP is ​

Answer 1

Answer:

Let the three positive numbers be a,ar,ar

2

which form an increasing G.P.

Also,a,2ar,ar

2

are in A.P.

⇒2ar−a=ar

2

−2ar

⇒4ar=ar

2

+a

⇒4r=r

2

+1

⇒r

2

−4r+1=0

Using quadratic formula, we get

r=

2(1)

4±

(−4)

2

−4(1)(1)

⇒r=

2

4±

12

⇒r=

2

4±2

3

⇒r=2±

3

But, we know that G.P is increasing

Therefore, r=2+

3

.

Answer 2

Answer:Let a, ar, ar2 be the numbers in GP.

2(2ar) = a + ar2

4r = 1+r2

r2-4r+1 = 0

Solving above equation using quadratic formula, we get

r = (4 ± √(16 – 4))/2

= (4 ± √12)/2

= 2 ± √3

Since it is increasing GP, r = 2 + √3

hope it will helpfull