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If (1+x)^n=sum(r=0)^n Crx^r then prove t...

If `(1+x)^n=sum_(r=0)^n C_rx^r` then prove that `sum_(r=0)^n (C_r)/((r+1)2^(r+1))=(3^(n+1)-2^(n+1))/((n+1)2^(n+1))`

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A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. Find the sum sum(k=1) (""^nC(2k-1))/(2k)

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

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

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

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

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

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

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

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

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

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

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  15. 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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  16. Find the sum sumsum(0lt=i<jlt=n)^n Ci^n Cj

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  17. If (1 + x + x^(2) + x^(3))^(n) = a(0) + a(1)x + a(2)x^(2)+"……….."a(3n)...

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  18. The coefficient of a^4b^5 in the expansion of (a+b)^9 is .

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  19. The coefficient in the third term of the expansion of (x^2-1/4)^n when...

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  20. Which is larger : (99^(50)+100^(50)) or (101)^(50).

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