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If a is the A.M. of ba n dc and the two ...

If `a` is the A.M. of `ba n dc` and the two geometric means are `G_1a n dG_2` , then prove that `G_1^3+G_2 ^3=2a bc`

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a is the A.M of b and `c => 2a = b + c`
Given, `G_1 `and `G_2` are G.Ms between b and c ` => b, G_1, G_2, c` are in G.P.
Let `r `be the common ratio of the G.P.
`⇒ G_1 = br,G_2 = br^2, c = br^3 `
...
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RD SHARMA-GEOMETRIC PROGRESSIONS-Solved Examples And Exercises
  1. Find two positive numbers whose difference is 12 an whose A.M. exce...

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  2. If one geometric mean G and two arithmetic means A1a n dA2 be inserted...

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  3. If a is the A.M. of ba n dc and the two geometric means are G1a n dG2 ...

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  4. If x in R , find the minimum value of the expression 3^x+3^(1-x)dot

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  5. If x ,y ,z are distinct positive numbers, then prove that (x+y)(y+z)(z...

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  6. If a ,b ,c are in G.P. then prove that (a^2+a b+b^2)/(b c+c a+a b)=(b+...

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  7. If 1/(a+b),1/(2b),1/(b+c) are three consecutive terms of an A.P., prov...

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  8. If a ,b ,c ,d are in G.P. prove that: (a b-c d)/(b^2-c^2)=(a+c)/b

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

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  10. The sum of three numbers a ,b ,c in A.P. is 18 . If a and b are each i...

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  11. If a ,b ,c are in G.P., prove that: 1) a(b^2+c^2)=c(a^2+b^2) , 2)...

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  12. If mth, nth and pth terms of a G.P. form three consecutive terms of a ...

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  13. If a ,b ,c are in G.P. and x ,y are the arithmetic means of a ,ba n db...

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  14. If x^a=x^(b//2)z^(b//2)=z^c , then prove that 1/a ,1/b ,1/c are in A.P...

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  15. If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , t...

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  16. If the sum of three numbers in G.P. is 38 and their product is 1728, ...

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  17. If the p^(t h) and q^(t h) terms of a G.P. are q and p respectively, s...

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  18. In a finite G.P. the product of the terms equidistant from the begi...

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  19. The product of first three terms of a G.P. is 1000. If 6 added to its ...

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  20. Find three numbers in G.P. whose sum is 52 and the sum of whose pro...

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