Given, sn=1+q+q^2++q^n ,Sn=1+(q+1)/2+((q...

Given, `s_n=1+q+q^2++q^n ,S_n=1+(q+1)/2+((q+1)/2)^2++((q+1)/2)^n ,q!=1` prove that `^n+1C_1+^(n+1)C_2s_1+^(n+1)C_3s_2++^(n+1)C_(n+1)s_n2^n S_ndot`

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Given,s_(n)=1+q+q^(2)+....+q^(n),S_(n)=1+(q+1)/(2)+((q+1)/(2))^(2)+...+((q+1)/(2))^(n),q!=1 prove that ^(n+1)C_(1)+^(n+1)C_(2)s_(1)+^(n+1)C_(3)s_(2)+......+^(n+1)C_(n+1)s_(n)=2^(n)S_(n)

Let S_k=1+q+q^2+...+q^k and T_k=1+(q+1)/2+((q+1)/2)^2+...+((q+1)/2)^k q!=1 then prove that sum_(r=1)^(n+1) ^(n+1)C_rS_(r-1)=2^ nT_n

Given s=1+q+q^(2)+...+q^(n),S_(n)=1+(q+(1)/(2))+(q+(1)/(2))^(2)+......+(q+(1)/(2))^(n) then prove that ^(n+1)C_(1)+^(n+1)C_(2)s_(1)+......,+^(n+1)C_(n+1)s_(n)=2^(n)s_(n)

Prove that (^(2n)C_0)^2+(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot

Let S_(n)=1+q+q^(2)+?+q^(n) and T_(n)=1+((q+1)/(2))+((q+1)/(2))^(2)+?+((q+1)/(2)) If alpha T_(100)=^(101)C_(1)+^(101)C_(2)xS_(1)+^(101)C_(101)xS_(100), then the value of alpha is equal to (A) 2^(99)(B)2^(101)(C)2^(100) (D) -2^(100)

Delta[[ Prove that ,, 11,1,1nC1,(n+1)C1,(n+2)C1(n+1)C2,(n+2)C2,(n+3)C2]]=1

Let S_n=1/1^2 + 1/2^2 + 1/3^2 +….. + 1/n^2 and T_n=2 -1/n , then :

If (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a+.^(n)C_(2)a^(2)+ . . +.^(n)C_(n)a^(n) , then prove that .^(n)C_(1)+2.^(n)C_(2)+3.^(n)3C_(3)+ . . .+n.^(n)C_(n)=n.2^(n-1) .

Prove that C_0+(C_1)/(2)+(C_2)/(3)+....+(C_n)/(n+1)=(2^(n+1)-1)/(n+1)

Consider (1+x)^(n+1)=A_(0)+A_(1)x+A_(2)x^(2)+………………+A_(n+1)x^(n+1) and a_(n)=1++q^(2)+……………+q^(n) and b_(n)=1+((q+1)/2)+((q+1)/2)^(2)+……….+((q+1)/2)^(n) Where q!=1 The value of (1-q)(A_(1)+A_(2)a_(1)) at n=1 is

A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise

Use a combinatorial argument to prove that (C(n,1))^2+2(C(n,2))^2+3(C...
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Prove that (""^nC1)/(""^nC0)+2*(""^nC2)/(""^nC1)+3*(""^nC3)/(""^nC2)+....
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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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Find the value of sum(p=1)^n(sum(m=p)^n^n Cm^m Cp)dot And hence, find ...
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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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Find the sum sumsum(0lt=i<jlt=n)^n Ci^n Cj
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If (1 + x + x^(2) + x^(3))^(n) = a(0) + a(1)x + a(2)x^(2)+"……….."a(3n)...
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The coefficient of a^4b^5 in the expansion of (a+b)^9 is .
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The coefficient in the third term of the expansion of (x^2-1/4)^n when...
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Which is larger : (99^(50)+100^(50)) or (101)^(50).
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The sum of the coefficeints of the polynominal (1 + x - 3x^(2))^(2163)...
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The sum of the numerical coefficients in the expansion of (2x+3y)^10 i...
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If the fifth term of the expansion (a^(2//3)+a^(-1))^n does not contai...
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The sum of the rational terms in the expansion of (sqrt(2)+ root(5...
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Write the middle term in the expansion of (x+1/x)^(10)dot
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If the coefficients of 2nd, 3rd and 4th terms in the expansion of(1+x)...
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If x^r occurs in the expansion of (x+1/x)^n then its coefficient is .
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If the sum of the coefficients in the expansion of (a+b)^n is 4096, th...
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The number of terms in the expansion of (1+x^(1/5))^55 which are free ...
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If n is even then the coefficient of x in the expansion of (1+x)^n*(1-...
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