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If G is the geometric mean of xa n dy th...

If `G` is the geometric mean of `xa n dy` then prove that `1/(G^2-x^2)+1/(G^2-y^2)=1/(G^2)`

Text Solution

Verified by Experts

As given `G=sqrt(xy)`
`therefore1/(G^(2)-x^(2))+1/(G^(2)-y^(2))=1/(xy-x^(2))+1/(xy-y^(2))`
`=1/(x-y){-1/x+1/y}`
`=1/(xy)=1/(G^(2))`
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CENGAGE-PROGRESSION AND SERIES-Examples
  1. Three non-zero numbers a ,b ,a n dc are in A.P. Increasing a by 1 or i...

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  2. If a, b, c are in A.P., b, c, d are in G.P. and 1/c ,1/d ,1/eare in A....

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  3. If G is the geometric mean of xa n dy then prove that 1/(G^2-x^2)+1/(G...

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  4. Insert four G.M.s between 2 and 486.

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  5. If A.M. and G.M. between two numbers is in the ratio m : n then prove ...

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  6. If a be one A.M and G1 and G2 be then geometric means between b and c ...

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  7. Determine the number of terms in G.P. <<an>>,ifa1=3,an=96a n dSn=189.

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  8. if S is the sum , P the product and R the sum of reciprocals of n term...

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  9. Find the sum to n terms of the sequence (x+1//x)^2,(x^2+1//x)^2,(x^3+1...

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  10. Prove that the sum to n terms of the series 11+103+1005+ i s(10//9)(10...

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  11. Find the sum of the following series up to n terms: (i) 5" "+" "55" "+...

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  12. Find the sum 1+(1+2)+(1+2+2^(2))+(1+2+2^(2)+2^(3))+ …. To n terms.

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  13. If the sum of the n terms of a G.P. is (3^(n)-1), then find the sum of...

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  14. Prove that in a sequence of numbers 49,4489,444889,44448889 in which e...

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  15. If f is a function satisfying f(x+y)=f(x)xxf(y) for all x ,y in N suc...

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  16. Using the sum of G.P., prove that a^(n)+b^(n)(a,binN) is divisble by a...

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  17. Sum the following geometric series to infinity: (sqrt(2)+1)+1+(sqrt(2...

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  18. The sum of infinite number of terms in G.P. is 20 and the sum of their...

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  19. If each term of an infinite G.P. is twice the sum of the terms foll...

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  20. If x=a+a/r+a/(r^2)+oo,y=b-b/r+b/(r^2)+oo,a n dz=c+c/(r^2)+c/(r^4)+oo, ...

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