[" 6.Wse prineiple of "M.I" .to prove that: "],[x+4x+7x+...---+(3n-2)x=(1)/(2)n(3n-1)x]

Using mathematical induction prove that x+4x+7x+......+(3n-2)x=(1)/(2)n(3n-1)x

By using the Principle of Mathematical Induction, prove the following for all n in N : x+4x+7x+......+(3n-2)x=1/2n (3n-1) x .

If x_(1),x_(2),x_(3)...x_(n) are in H.P,then prove that x_(1)x_(2)+x_(2)x_(3)+xx x-3x_(4)+...+x_(n-1)x_(n)=(n-1)x_(1)x_(n)

If x^(p) occurs in the expansion of (x^(2)+1/x)^(2n) prove that its coefficient is ((2n)!)/([(1)/(3)(4n-p)]![(1)/(3)(2n+p)]!)

If x_1,x_2,x_3….,x_n are in H.P. prove that x_1x_2+x_2x_3+x_3x_4+……….+x_(n-1)x_n=(n-1)x_1x_n

If x_1,x_2,x_3….,x_n are in H.P. prove that x_1x_2+x_2x_3+x_3x_4+……….+x_(n-1)x_n=(n-1)x_1x_n

If S_(n)=(x+y)+(x^(2)+xy+y^(2))+(x^(3)+x^(2)y+y^(2)x+y^(3))+…n terms then prove that (x-y)S_(n)=[(x^(2)(x^(n)-1))/(x-1)-(y^(2)y^(n)-1)/(y-1)] .

If (1+x+x^(2))^(n)=a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)+.....+a_(2n)x^(2n)" prove that",a_(0)+a_(2)+a_(4)+.......a_(2n)=(1)/(2)(3^(n)+1) .

Prove that the term independent of x in the expansion of (x+(1)/(x))^(2n) is (1.3.5....(2n-1))/(n!)*2^(n)

Prove that the term independent of x in the expansin of (x+(1)/(x))^(2n) is (1.3.5(2n-1))/(n!)*2^(n)