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3C0+3^2(C1)/2+3^3(C2)/3+.............3^(...

`3C_0+3^2(C_1)/2+3^3(C_2)/3+.............3^(n+1)*(C_n)/(n+1)` eqaul to

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3C_0+3^2(C_1)/2+3^3(C_2)/3+.............3^(n+1)*(C_n)/(n+1) equal to

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

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

k. C_0 + k^2 . (C_1)/(2)+k^3. (C_2)/(3)+…..+ k^(n+1). (C_n)/(n+1)=

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If (1+x)^n=C_0+C_1x+C_2x^2+……..+C_nx^n in N prove that (a) 3 C_0- 8C_1+13C_2-18C_3+...."upto" (n+1) term=0 if n ge 2 (b ) 2C_0+2^2(C_1)/(2)+2^3(C_2)/(3)+2^4C_(3)/4+....+2^(n+1)(C_n)/(n+1)=(3^n+1-1)/(n+1) ( c) C_0^2+(C_1^2)/2+C_2^2/3+....+C_n^2/(n+1)=((2n+1)!)/(((n+1)!)^2)

(C_0)/(1. 3)-(C_1)/(2. 3)+(C_2)/(3. 3)-(C_3)/(4. 3)+...... +(-1)^n(C_n)/((n+1)*3) is

(C_0)/(1. 3)-(C_1)/(2. 3)+(C_2)/(3. 3)-(C_3)/(4. 3)+...... +(-1)^n(C_n)/((n+1)*3) is

C_1/C_0+2C_2/C_1+3C_3/C_2+............+nC_n/C_(n-1)=(n(n+1))/2