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If n is a positive integer, then prove t...

If n is a positive integer, then prove that `81^n + 20 n - 1 `is divisible by 100`

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AAKASH SERIES-BINOMIAL THEOREM-SOLVED EXAMPLES
  1. If the 2^("nd"),3^("rd")" and "4^("th") terms in the expansion of (a+x...

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  2. If the coefficients of x^(9),x^(10),x^(11) in expansion of (1+x)^(n) a...

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  3. If n is a positive integer, then prove that 81^n + 20 n - 1 is divisib...

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  4. Using binomial theorem, prove that 5^(4n)+52n-1 is divisible by 676 fo...

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  5. 3. C0 + 7. C1 + 11. C2 +…...+ (4n+3) . Cn=

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  6. With usual notations find the value of 3.C1 + 5.C2 + 7.C3 + ……….+n ter...

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  7. With usual notations prove that C1+2.C2x + 3.C3x^2 + ……+2n.C(2n) ...

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  8. Prove that (""^20C1)/(""^20C0) + 2. (""^20C2)/(""^20C1) + 3. (""^2...

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  9. If n is a positive integer and C4 = ""^nCr then find the value of sum(...

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  10. Prove that sum(r = 1)^(n+1) (2^(r +1) C(r - 1) )/(r (r + 1)) = (3^(n+...

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  11. Prove that sum(r = 0)^n r^2 . Cr = n (n +1).2^(n-2)

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  12. Prove that sum(r = 0)^n r^3 . Cr = n^2 (n +3).2^(n-3)

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  13. If Cr denotes ""^nCr then show that C0 + (C1)/(2) + (C2)/(3) x^2 +...

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  14. If C0, C1, C2 ,……..Cn are the coefficient in the expansion of (1 + x)^...

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  15. With usual notations prove that C0 + 3.C1 + 3^2.C2 + ………..+3^n .Cn...

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  16. With usual notations prove that C1/C0 + 2. C2/C1 + 3.C3/C2 + ……+n.(...

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  17. Prove that (C0 + C1) (C1 + C2) …..(C(n-1) + Cn) = ((n+1)^n)/(n!) (C1.C...

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  18. Prove that sum(r = 1)^n r^3 ((Cr)/(C(r - 1))^2) = (n (n + 1)^2 (n+2))/...

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  19. Prove that following C(0)+(3)/(2).C(1)+(9)/(3).C(2)+(27)/(4).C(3)+……...

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  20. Find the range of x for which the binomial expansion of the following ...

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