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I have a doubt, kindly clarify. If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

 

  • Option 1)

    \frac{8}{5}

  • Option 2)

    \frac{4}{3}

  • Option 3)

    1

  • Option 4)

    \frac{7}{4}

 
Answers (2)
199 Views
N neha

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, - - - - - - -

r = 2

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

r = 1/10

 


 {A}, {B},{C} are in  G.P where {A}, {B},{C}  are Terms of an A.P

 

Let first term is a and common difference is d, and common ratio be r then

A = a+d

B = a+4d

C= a+8d

\therefore \: \frac{B}{A}=\frac{C}{B}=\frac{r}{1}

\frac{a+4d}{a+d} =\frac{a+8d}{a+4d}=\frac{r}{1}

\therefore \frac{a+4d+a+d}{a+4d-a-d}=\frac{r+1}{r-1}

\Rightarrow \frac{2a+5d}{3d}=\frac{r+1}{r-1}-----(i)

    \frac{a+8d+a+4d}{a+8d-a-4d} =\frac{r+1}{r-1}

\Rightarrow \frac{2a+12d}{4d}=\frac{r+1}{r-1}------(ii)

from (i) and (ii)

\frac{2a+5d}{2a+12d}=\frac{3}{4}

8a+20d=6a+36d

2a=16d

a=8d

\Rightarrow \frac{r+1}{r-1}=\frac{2\times8d+5d}{3d}=\frac{16d+5d}{3d}=\frac{21}{3}=7

\therefore r+1=7r-7

8=6r

r=\frac{4}{3}


Option 1)

\frac{8}{5}

Incorrect option

Option 2)

\frac{4}{3}

Correct option

Option 3)

1

Incorrect option

Option 4)

\frac{7}{4}

Incorrect option

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