Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is :

  • Option 1)

    2-\sqrt{3}

  • Option 2)

    2+\sqrt{3}

  • Option 3)

    \sqrt{2}+\sqrt{3}

  • Option 4)

    3+\sqrt{2}

 

Answers (2)

As we learnt in 

Common ratio of a GP (r) -

The ratio of two consecutive terms of a GP

- wherein

eg: in 2, 4, 8, 16, - - - - - - -

4r=1+r^{2}

r = 2

and in 100, 10, 1, 1/10 - - - - - - -

r = 1/10

 

 and

 

Arithmetic mean of two numbers (AM) -

A=\frac{a+b}{2}

- wherein

It is to be noted that the sequence a, A, b, is in AP where, a and b are the two numbers.

 

a, ar, ar2 in G.P

a, 2ar, ar2 in A.P (given)

\therefore 2\times 2ar= a+ar^{2}

    4r=1+r^{2}

\therefore r^{2}=4r+1=0

\therefore r=\frac{4+\sqrt{16-4}}{2}

      =\frac{4+\sqrt{12}}{2}

      =2+\sqrt{3}

 

 


Option 1)

2-\sqrt{3}

This option is incorrect

Option 2)

2+\sqrt{3}

This option is correct

Option 3)

\sqrt{2}+\sqrt{3}

This option is incorrect

Option 4)

3+\sqrt{2}

This option is incorrect

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