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For ninN, prove that ((n+1)/2)^ngt n!...

For `ninN`, prove that `((n+1)/2)^ngt` n!

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PATHFINDER-BINOMIAL THEOREM AND PRINCIPLE OF MATHEMATICAL INDUCTION-QUESTION BANK
  1. If coefficient of x^2 and x^11 are 27 and -192 respectively of (1+ax+2...

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  2. Find the coefficient of x^5 in the expansion of (1+x)^21+(1+x)^22+...+...

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  3. Determine the x-independent term in the expansion of (1+4x)^p(1+1/(4x)...

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  4. For ninN , 2^(3n)-1 is divisible by

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  5. For ninN , n^3+2n is divisible by

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  6. ForninN , 3^(2n-1)+2^(n+1) is always divisible by

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  7. For ninN 2^(3n)-7n-1 is always divisible by

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  8. The greatest positive integer divides (n+1)(n+2)..........(n+r) is

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  9. Applying the principle of mathematical induction (P.M.I.) prove that 1...

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  10. Using mathematical induction show 7+77+777+......+n terms =7/81(10^(n+...

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  11. Applying P.M.I. prove that x^n-y^n is always divisible by x+y where n...

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  12. Applying the principle mathematical induction (P.M.I.) show that 5^(2n...

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  13. Applying P.M.I. prove that (1+x)^n gt 1+nx where n is a pos integer an...

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  14. Prove that (costheta+isintheta)^n=cosntheta+isinntheta by P.M.I. where...

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  15. For which natural numbers n the inequality 2^ngt2n+1is true?

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  16. For n being a natural number prove that 1.1!+2.2!+3.3!+.....+n.n! =(n+...

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  17. For ninN, prove that ((n+1)/2)^ngt n!

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  18. Show that 101^(50)gt 99^(50)+100^(50)

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  19. Using P.M.I. prove that 2^ngt n for all nge1:ninN

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  20. If nge3 is an integer prove that 2n+1 lt 2^n by P.M.I.

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