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1 week, 6 days ago

Need clarity, kindly explain! - Sequence and series - JEE Main

 Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of  \frac{1}{a}  and \frac{1}{b}

if \frac{1}{M}:G  is 4:5 then a:b can be:

  • Option 1)

    1:4

  • Option 2)

    1:2

  • Option 3)

    2:3

  • Option 4)

    3:4

 
Answers (1)
38 Views
S solutionqc
Answered 1 week, 6 days ago

As we learnt in 

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.

and

Geometric mean of two numbers (GM) -

GM= \sqrt{ab}

- wherein

It is to be noted that a,G,b are in GP and a,b are two non - zero numbers.

 

Given G= ab

2M=\frac{1}{a}+\frac{1}{b}

and  \frac{\frac{1}{M}}{G}=\frac{4}{5}

\therefore GM=\frac{5}{4}

\therefore G^{2}M^{2}=\frac{25}{16}

               =ab\times \left (\frac{a+b}{2ab} \right )^{2}= \frac{25}{16}

               =\frac{ab\times \left ( a+b \right )^{2}}{4a^{2}b^{2}}= \frac{25}{16}

\therefore 4a^{2}+4b^{2}-17ab=0

\therefore 4\left ( \frac{a}{b} \right )^{2}-17\left ( \frac{a}{b} \right )+4=0

\therefore \frac{a}{b}=1:4


Option 1)

1:4

Option 2)

1:2

Option 3)

2:3

Option 4)

3:4

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