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If a,b,c are in G.P. and a^(1/x)=b^(1/y)...

If `a,b,c` are in `G.P.` and `a^(1/x)=b^(1/y)=c^(1/z)` prove that `x, y, z` are in `A.P.`

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Given, `a,b,c` are in `G.P.`

`a^(1/x)=b^(1/y)=c^(1/z)`` \ \ \ (1)`

Geometric mean relation:

`b^2=ac`` \ \ \ (2)`

Taking log in eq `(1)`

`loga^(1/x)=logb^(1/y)=logc^(1/z)`

`(loga)/x=(logb)/y=(logc)/z=k`

`log a=kx implies a=e^(kx)`

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RD SHARMA-GEOMETRIC PROGRESSIONS-Solved Examples And Exercises
  1. If a ,\ b ,\ c ,\ d are in G.P., show that: (a b+b c+c d)^2=(a^2+b^2+c...

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  2. If a ,b ,c,d are in G.P. prove that (a^n+b^n),(b^n+c^n),(c^n+d^n) are...

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  3. If a,b,c are in G.P. and a^(1/x)=b^(1/y)=c^(1/z) prove that x, y, z ar...

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  4. If a^2+b^2,a b+b c and b^2+c^2 are in G.P., then a ,b ,c are in

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  5. If a , b , c are in G.P. prove that loga , logb , logc are in A.P.

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  6. If a ,\ b ,\ c are in G.P. prove that 1/((log)a m),1/((log)b m),1/((lo...

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  7. Find k such that k+9, k-6 and 4 from three consecutive terms of a G.P.

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  8. Three numbers are in A.P. and their sum is 15. If 1,3,9 be added to th...

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  9. The sum of three numbers which are consecutive terms of an A.P. is 21....

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  10. The sum of three numbers in G.P. is 56. If we subtract 1,7,21 from the...

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  11. If a ,\ b\ c are in G.P., prove that the following are also in G.P.: a...

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  12. If a , b, c are in G.P., prove that the following are also in G.P.: a^...

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  13. If a, b, c are in G. P., prove that a^2+ b^2, ab + bc, b^2 +c^2 are al...

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  14. If (a-b),\ (b-c),(c-a) are in G.P. then prove that (a+b+c)^2=3(a b+b c...

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  15. If the 4th, 10th and 16th terms of a G.P. are x, y, and z respectivel...

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  16. If a, b, c are in A.P and a, b, d are in G.P, prove that a, a -b, d ...

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  17. If the pth, qth, rth, and sth terms of an A.P. are in G.P., then p-q ,...

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  18. If a ,\ b ,\ c are in A.P. b ,\ c ,\ d are in G.P. and 1/c ,1/d ,1/e a...

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  19. If a, b, c are in A.P and a, b, d are in G.P, prove that a, a -b, d ...

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  20. If p^(t h),\ q^(t h),\ a n d\ r^(t h) terms of an A.P. and G.P. are bo...

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