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  • Can someone help me with this, 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 :

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

Posted by

Sabhrant Ambastha

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