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Prove that ""^nC0-2*""^nC1+3*""^nC2-...+...

Prove that `""^nC_0-2*""^nC_1+3*""^nC_2-...+(-1)""^n(n+1)""^nC_n=0`

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

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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)

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}

Prove that : (1+""^nC_1+""^nC_2+""^nC_3+...+""^nC_n)^2=1+""^(2n)C_1+""^(2n)C_2+""^(2n)C_3+...+""^(2n)C_(2n)

Statement - 1: The value of ((20),(0))((20),(1))-((20),(1))((20),(9))+((20),(2))((20),(8))-((20),(3))((20),(7))+...+((20),(10))((20),(0))=0, where ((n),(r))=^nC_r Statement - 2: .^nC_0-^nC_1+^nC_2-^nC_3+...+(-1)^n .^nC_n=0.

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

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A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. Prove that ""^nC0+2*""^nC1+3*""^nC2+...+(n+1)""^nCn=(n+2)2^(n-1)

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

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  3. Prove that ""^nC0-2*""^nC1+3*""^nC2-...+(-1)""^n(n+1)""^nCn=0

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  4. If sn=""^nC0+2*""^nC1+3*""^nC2+...+(n+1)*""^nCn then find sum(n=1)^oos...

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  5. Find the sum :1*""^nC0+2*""^nC1+3*""^nC2+4*""^nC3+…, where n is an odd...

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  6. Show that :""^nC0*m-""^nC1*(m-1)+""^nC2*(m-2)-...+(-1)^n*""^nCn*(m-n)=...

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

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

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

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

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

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

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

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

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

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

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

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

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