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Prove by binomial expansion that sum(k=1...

Prove by binomial expansion that `sum_(k=1)^nk^2*"^nC_k=n(n+1)2^(n-2)`

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A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. Show that :""^nC0*m-""^nC1*(m-1)+""^nC2*(m-2)-...+(-1)^n*""^nCn*(m-n)=...

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  2. Evaluate sum(r=1)^npr/r*"^nCr where pr denotes the sum of the first r ...

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  3. Prove by binomial expansion that sum(k=1)^nk^2*"^nCk=n(n+1)2^(n-2)

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  4. Evaluate sum(r=0)^n(r+1)^2*"^nCr

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  5. If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n then prove that 2.C0+2^2C1/2+2...

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  6. If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n then prove that C0-1/2C1+1/3C2...

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  7. Prove that 3*""^10C0+3^2*(""^10C1)/2+3^3*(""^10C2)/3+...3^11*(""^10C10...

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  8. Prove that 2*""^nC0+2^2*(""^nC1)/2+2^3*(""^nC2)/3+...2^(n+1)*(""^nCn)/...

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  9. Find the sum sum(k=0)^n("^nCk)/((k+1)(k+2))

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  10. Find the sum sum(r=0)^n(-1)^r*(""^nCr)/(""^(r+3)Cr)

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  11. Find the sum sum(k=1) (""^nC(2k-1))/(2k)

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  12. Find the sum sum(k=0)^n ("^nCk)/(k+1)

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  13. If (1+x)^n=sum(r=0)^n Crx^r then prove that sum(r=0)^n (Cr)/((r+1)2^(r...

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  14. Find sum(r=0)^n(r+1)*"^nCrx^r

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  15. Show that C0^2-C1^2+C2^2-C3^2+...........+(-1)^n Cn^2=0 or (-1)^(n/2)C...

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  16. Prove that (""^(2n)C(0))^(2)-(""^(2n)C(1))^(2)+(""^(2n)C(2))^(2)-…+(""...

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  17. Sum of the products of the binomial coefficients C0,C1,C2,......Cn ta...

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  18. Find the sum ^20C10.^15C0+^20C9.^15C1+^20C8.^15C2+....+^20C0.^15C10

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  19. Prove that sum(r=1)^k (-3)^(r-1) "^(3n)C(2r-1) =0 , where k=(3n)/2 and...

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  20. If p+q=1, then show that sum(r=0)^n r^2^n Crp^r q^(n-r)=n p q+n^2p^2do...

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