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Show that C(0) +(C(0) +C(1))+(C(0)+C(1)+...

Show that `C_(0) +(C_(0) +C_(1))+(C_(0)+C_(1)+C_(2)) +…+(C_(0) +C_(1)+…+C_(n)) =(n+2).2^(n-1)`

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If (C_0 + C_1) (C_1+C_2)…(C_(n-1) + C_n)=k C_0 C_1 C_2… C_n then k=

C_0 C_1+C_1 C_2 + C_2 C_3+…+ C_(n-1) C_n=

Prove that : Prove that (C_(1))/(C_(0))+2.(C_(2))/(C_(1))+3.(C_(3))/(C_(2))+….+n.(C_(n))/(C_(n-1))=(n(n+1))/(2)

If C_0, C_1, C_2 ,……..C_n are the coefficient in the expansion of (1 + x)^n then show that C_0 C_r + C_1 C_(r + 1) + C_2 C_(r + 2) + ………..+ C_(n-r).C_n = ((2n)!)/((n-r)!(n+r)!)

Prove that (C_(0)+C_(1))(C_(1)+C_(2))(C_(2)+C_(3))....(C_(n-1)+C_(n))=((n+1)^(n))/(n!).C_(0).C_(1).C_(2).....C_(n).

Prove that C_(0)+3.C_(1)+5.C_(2)+….+(2n+1).C_(n)=(2n+2).2^(n-1).

Prove that : For n = 0, 1, 2, 3, ………., n, prove that C_(0).C_(r)+C_(1).C_(r+1)+C_(2).C_(r+2)+….+C_(n-r).C_(n) =""^(2n)C_((n+r)) and hence deduce that C_(0).C_(1)+C_(1).C_(2)+C_(2).C_(3)+……..+C_(n-1).C_(n)=""^(2n)C_(n+1)

Prove that : For n = 0, 1, 2, 3, ………., n, prove that C_(0).C_(r)+C_(1).C_(r+1)+C_(2).C_(r+2)+….+C_(n-r).C_(n) =""^(2n)C_((n+r)) and hence deduce that Prove that : C_(0)^(2)+C_(1)^(2)+C_(2)^(2)+….+C_(n)^(2)=""^(2n)C_(n)

Prove that C _(0) +2 . C _(1) + 2 ^(2) . C _(2) + .......+ 2 ^(n) . C _(n) = 3 ^(n).

((C_0 + C_1)(C_1 + C_2)(C_2 + C_3)………(C_(n-1) + C_n) )/(C_0C_1C_2…C_n)

AAKASH SERIES-BINOMIAL THEOREM-EXERCISE - 1.4 (Level-2)
  1. Prove that C(0)-(1)/(3)*C(1)+(1)/(5)*C(2) - …+(-1)^(n)*(1)/(2n+1)C(n) ...

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  2. If a, b, c, d be 3^(rd), 4^(th), 5^(th) and 6^(th) terms of the expans...

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  3. Show that sum(r=0)^(n) (-1)^(r ). ""^(n)C(r ) {(1)/(2^(r )) +(3^(r ))/...

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  4. If sum(r = 0)^(2n) a(r ) (x-2)^(r ) = sum(r = 0)^(2n) b(r ) (x-3)^(r )...

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  5. Show that the coefficient of x^(n)y^(n) in the expansion of {(1+x)(1+...

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  6. If (1)/(1!(n-1)!) + (1)/(3!(n-3)!) + (1)/(5!(n-5)!) +…. =

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  7. If sum(r =0)^(n) ((r+2)/(r+1)) *""^(n)C(r ) =(255)/(6). Find n.

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  8. Show that 1sum(r =0)^(50) (""^(100)C( r))*(""^(200)C(150+r ))=""^(300)...

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  9. Prove that sum(k=1)^(n-r ) ""^(n-k)C(r )= ""^(n)C( r+1) .

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  10. Show that C(0) +(C(0) +C(1))+(C(0)+C(1)+C(2)) +…+(C(0) +C(1)+…+C(n)) =...

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  11. If x > 0 , write the first negative term in the expansion of (1 + 2x)^...

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  12. If x is small and the expansion of (a)/((1-x)^(2)) +(b)/((2+3x)^(2)) ...

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  13. If x is small and if the expansion of a + (b)/(1 + 2x) + ( c)/(1 -3x^2...

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  14. If (1+2x)/(1-x-x^(2)) =a(0)+a(1)x +a(2)x^(2) +a(3)x^(3)+… then find (a...

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  15. If x is nearly equal to 1 show that (a) (px^(p)-qx^(q))/(p-q)=x^(p+q...

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  16. If |x| lt 1 then 1+n ((2x)/(1+x))+(n(n+1))/(2!) ((2x)/(1=x))^2 + ….......

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  17. Prove the following sum(i=1)^(n) sum(j=1)^(n) sum(k=1)^(n) sum(l=1)...

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  18. Find the value of sum(i=1)^n sum(i=1)^n sum(k=1)^n k

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  19. Prove the following sum(1le i) sum(lt j le n) (i+j)=(n(n^(2)-1))/(2...

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  20. Prove the following sum(r=0)^(n)sum(s=0)^(n)(C(r )*C(s))=4^(n)

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