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If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n ...

If `(1+x)^n=C_0+C_1x+C_2x^2+C_3x^3+...+C_nx^n` then prove that `C_0-1/2C_1+1/3C_2-1/4C_3+...+(-1)^n*C_n/(n+1)=1/(n+1)`

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A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. Evaluate sum(r=0)^n(r+1)^2*"^nCr

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

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

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

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

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

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

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

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

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

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

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

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

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

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  17. 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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  18. Use a combinatorial argument to prove that (C(n,1))^2+2(C(n,2))^2+3(C...

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  19. Prove that (""^nC1)/(""^nC0)+2*(""^nC2)/(""^nC1)+3*(""^nC3)/(""^nC2)+....

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  20. Given, sn=1+q+q^2++q^n ,Sn=1+(q+1)/2+((q+1)/2)^2++((q+1)/2)^n ,q!=1 p...

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