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

Prove that `2*""^nC_0+2^2*(""^nC_1)/2+2^3*(""^nC_2)/3+...2^(n+1)*(""^nC_n)/(n+1)=(3^(n+1)-1)/(n+1)`

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Prove that ""^nC_0+2*""^nC_1+3*""^nC_2+...+(n+1)""^nC_n=(n+2)2^(n-1)

Prove that ""^nC_0+3*""^nC_1+5*""^nC_2+...+(2n+1)""^nC_n=(n+1)2^(n)

Show that : ""^nC_0*m-""^nC_1*(m-1)+""^nC_2*(m-2)-...+(-1)^n*""^nC_n*(m-n)=0

Prove that (""^nC_1)/(""^nC_0)+2*(""^nC_2)/(""^nC_1)+3*(""^nC_3)/(""^nC_2)+...+n*(""^nC_n)/(""^nC_(n-1))=frac{n(n+1)}{2}

If s_n=""^nC_0+2*""^nC_1+3*""^nC_2+...+(n+1)*""^nC_n then find sum_(n=1)^oos_n .

Find the sum : 1/2*""^(n)C_0+""^(n)C_1+2*""^(n)C_2+2^2*""^nC_3+...+2^(n-1)*""^(n)C_n .

Prove that "^nC_r + 2. ^nC_(r-1) + ^nC_(r-2) = ^(n+2)C_r

Prove that (2nC_(1))^(2)+2.(2nC_(2))^(2)+3*(2nC_(3))^(2)+......+2n*(2nC_ (2n))^(2)=((4n-1)!)/((([(2n-1)!])^(2))

Prove that (nC_(0))/(1)+(^nC_(2))/(3)+(^nC_(4))/(5)+(^nC_(6))/(7)+...=(2^(n))/(n+1)

A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n then prove that C0-1/2C1+1/3C2...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  18. 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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  19. Find the value of sum(p=1)^n(sum(m=p)^n^n Cm^m Cp)dot And hence, find ...

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  20. The value of ^(2n+1)C0^2+^(2n+1)C1^2+^(2n+1)C2^2+....+^(2n+1)Cn^2 is e...

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