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(C0+C1)(C1+C2)(C2+C3)(C3+C4)...........(...

`(C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n)=(C_0C_1C_2.....C_(n-1)(n+1)^n)/(n!)`

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`LHS = (C_(0) + C_(1)) (C_(1)+ C_(2)) (C_(2) + C_(3))...(C_(n-1) + C_(n))`
`= prod_(r=1)^(n) (C_(r-1) + C_(r)) = prod_(r=1)^(n) (""^(r+1)C ) " " [ because ""^(n)C_(r) + ""^(n)C_(r-1) = ""^(n+1)C_(r)]`
`= prod_(r=1)^(n) ((n+1)/(r))""^(n)C_(r-1) " "[because ""^(n)C_(r) = (n)/(r) ""^(n-1)C_(r-1)]`
`= prod_(r=1)^(n)(n+1) . = prod_(r=1)^(n)(1)/(r).prod_(r=1)^(n)C_(r-1)`
`= (n+1)^(n) . (1)/(n!) (C_(0)C_(1) C_(2) ...C_(n-1))`
` = ((n+1)^(n))/(n!) (C_(0)C_(1) C_(2) ...C_(n-1))` = RHS
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(C_(0)+C_(1))(C_(1)+C_(2))(C_(2)+C_(3))(C_(3)+C_(4)).........(C_(n-1)+C_(n))=(C_(0)C_(1)C_(2).....C_(n-1)(n+1)^(n))/(n!)

( (C_0 +C_1)(C_1 + C_2)(C_2 +C_3).....(C_(n-1) +C_n))/(C_0C_1C_2....C_n)= (A) (n+1) ^n/n (B) (n+1)^(n-1)/n! (C) (n+1)^n/(n!)(D) (n+1)/(n!)

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

(C_(0))/(1)+(C_(1))/(2)+(C_(2))/(3)+.........(C_(n))/(n+1)=

(C_(1))/(C_(0))+2(C_(2))/(C_(1))+3(C_(3))/(C_(2))+.........+n(C_(n))/(C_(n-1))=(n(n+1))/(2)

C_(0)C_(1)+C_(1)C_(2)+C_(2)C_(3)+...+C_(n-1)C_(n)

C0-(C1)/(2)+(C2)/(3)-............+(-1)^(n)(Cn)/(n+1)=(1)/(n+1)

ARIHANT MATHS-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
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  14. In the expansion of ((x+1)/(x^(2/3)-x^(1/3)+1)-(x-1)/(x-x^(1/2)))^10 ...

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  15. The coefficients of three consecutive terms of (1+x)^(n+5) are in the ...

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  16. If the coefficients of x^(3) and x^(4) in the expansion of (1 + ax + b...

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  17. Coefficient of x^(11) in the expansion of (1+x^2)(1+x^3)^7(1+x^4)^(12)...

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  18. The sum of coefficients in integral powers of x in the binominal expan...

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  19. The coefficient of x^9 in the expansion of (1+x)(16 x^2)(1+x^3)(1+x^(1...

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  20. If the number of terms in the expansion of (1-2/x+4/(x^2))^n , x!=0, i...

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