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(b) Find the value of sum(r=m)^n .^rCm,n...

(b) Find the value of `sum_(r=m)^n .^rCm,n>m`

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A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. Find the coefficient of x^3 in the expansion of 1+(1+x)+(1+x)^2+(1+x)^...

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  2. Find the coefficients of x^(50) in the expression (1+x)^(1000)+2x(1+x)...

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  3. (b) Find the value of sum(r=m)^n .^rCm,n>m

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  4. Evaluate (""^3C3+""^4C3+""^5C3+...+""^nC3)xx(""^nC3+""^nC4+""^nC5+...+...

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  5. The value of ^n C1+^(n+1)C2+^(n+2)C3++^(n+m-1)Cm is equal to ^m+n C(n-...

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  6. Prove that ""^(n+1)C2+2*sum(k=2)^n""^kC2=sum(k=1)^nk^2

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  7. If (1+x)^n=C0+C1x+C2x^2+...+Cnx^n , find the sum of the following seri...

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

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

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

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

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

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

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

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

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

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

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

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

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