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If a ,b ,c are in G.P., prove that: 1) ...

If `a ,b ,c` are in G.P., prove that: 1) `a(b^2+c^2)=c(a^2+b^2)`
, 2) `A^2b^2c^2(1/(a^3)+1/(b^3)+1/(c^3))=a^3+b^3+c^3`
, 3) `((a+b+c)^2)/(a^2+b^2+c^2)=(a+b+c)/(a-b+c)`
, 4)`1/(a^2-b^2)+1/(b^2)=1/(b^2-c^2)`
,5) `(a+2b=2c)(a-2b+2c)=a^2+4c^2dot`

Text Solution

Verified by Experts

(i) `a(b^2 + c^2) = c(a^2 +b^2)`
Given that `a, b, c` are in GP.
By using the property of geometric mean, `b^2 = ac`
Let us consider LHS: `a(b^2 + c^2) `
...
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RD SHARMA-GEOMETRIC PROGRESSIONS-Solved Examples And Exercises
  1. 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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  2. The sum of three numbers a ,b ,c in A.P. is 18 . If a and b are each i...

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

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

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

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

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

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

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

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

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  13. Three numbers are in G.P. whose sum is 70. If the extremes be each ...

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  14. If the continued product of three numbers in G.P. is 216 and the sum o...

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  15. Find four numbers in G.P. whose sum is 85 and product is 4096.

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  16. Find three numbers in G.P. whose sum is 13 and the sum of whose square...

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  17. Find 4t h term from the end of the G.P. 3,6,12 ,24 , ,3072.

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  18. Which term of the G.P. 2,1/2,1/4, i s1/(128)?

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  19. Which term of the G.P. 5,10 ,20 ,40 , i s5120 ?

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  20. The fourth, seventh and the last term of a G.P. are 10 ,80a n d2560 re...

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