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If one geometric mean `G` and two arithmetic means `A_1a n dA_2` be inserted between two given quantities, prove that `G^2=(2A_1-A_2)(2A_2-A_1)dot`

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The correct Answer is:
`(2A_(1)-A_(2))(2A_(2)-A_(1))`
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MODERN PUBLICATION-SEQUENCES AND SERIES-ILLUSTRATIVE EXAMPLES
  1. Which term in the A.P. 5,2,-1,… is -22 ?

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  2. Which term of the sequence, 4, 3 5/7, 3 3/7 …………. Is the first negativ...

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  3. Which term of the sequence: 16-6i, 15-4i,14-2i………. Is pure imaginary?

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  4. Show that there is no A.P. which consists of only distinct prime nu...

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  5. The sum of first 12 terms of a G.P. is five times the sum of the first...

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  6. If 7 times the 7th term of an AP is equal to 11 times its 11th term, s...

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  7. Find the number of terms common to the two AP's 3,7,11,15.... 407 and...

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  8. If the pth, qth and rt terms of an A.P. be x,y,z respectively show tha...

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  9. Insert three numbers between 1 and 256 so that the resulting sequen...

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  10. The arithmetic mean between two numbers is 10 and their geometric mean...

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  11. The A.M. between two distinct positive numbers is twice the G.M. betwe...

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

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  13. Find all sequences which are simultaneously A.P. and G.P.

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  14. The sum of first three terms of a G.P. is (13)/(12)and their product ...

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

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  16. If p,q,r are in A.P. while x,y,z are in G.P., prove that x^(q-r)y^(r-p...

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  17. Find the sum to infinity of the series: 1+3/2+5/2^2+7/2^3+..........oo

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  18. If the sum to infinity of the series: 3+(3+d).1/4+(3+2d).1/(4^(2))+………...

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  19. Sum up the following series to n terms:3+7+14+24+37+…………..

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  20. Find the nth term and the sum of n term of the series 6+9+21+69+261...

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