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Prove that ((a+b)/2)^(a+b) le a^a.b^b.[a...

Prove that `((a+b)/2)^(a+b) le a^a.b^b.[a,binN]`.

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PATHFINDER-PROGRESSION AND SERIES -QUESTION BANK
  1. Prove that a^4+b^4+c^4>abc(a+b+c). [a,b,c are distinct positive real n...

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  2. Show that the greatest value of xyz(d-ax-by-cz) is d^4/(4^4abc).

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  3. Prove that ((a+b)/2)^(a+b) le a^a.b^b.[a,binN].

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  4. N arithmetic means are inserted in between x and 2y and then between 2...

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  5. Let Sn denote the sum upto n terms of an AP . If Sn=n^2P and Sm=m^2P ...

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  6. If s1 ,s2 and s3 are the sum of first n,2n,3n terms respectively of an...

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  7. Let a1,a2 ,a3 ...... be an A.P. Prove that sum(n=1)^(2m)(-1)^(n-1)an^2...

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  8. A G.P. consists of 2n terms . If the sum of the terms occupying the od...

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  9. If G1 ,G2 are geometric means , and A is the arithmetic mean between ...

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  10. Show that abs((a,b,aalpha+b),(b,c,balpha+c),(aalpha+b,balpha+c,0):}=0 ...

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  11. If Sn=1+1/2+1/2^2+........+1/2^(n-1) . Calculate the least value of n ...

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  12. Prove that the number of the sequence 121,12321,1234321,...... are eac...

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  13. Find the sum of n terms of series 1+5((4n+1)/(4n-3))+9((4n+1)/(4n-3))^...

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  14. Find the sum of the products of the integers 1,2,3,....n taken two at ...

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  15. Sum the series n+(n-1)x+(n-2)x^2+.......+2x^(n-2)+x^(n-1).

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  16. Find 1+2^2x+3^2x^2+4^2x^3+.... to infty absx lt 1.

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  17. Find the sum of 1st n terms of the sequence 3,6,15,42,123,...

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  18. Let Sn denote the sum of first n terms of the sequence 1,5,14,30,55,.....

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  19. Finds the sum of first n terms of the series 3/(1^2xx2^2)+5/(2^2xx3^2)...

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  20. How many terms of the series 54+51+48+45+ ... must be taken to make 51...

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