Show that `(1-a)^(3)+(b-1)^(3)+(a-b)^(3)=3(1-a)(b-1)(a-b)`

If a, b, c are in G.P., show that a^(2)b^(2)c^(2)((1)/(a^(3))+(1)/(b^(3))+(1)/(c^(3))) = a^(3) + b^(3) + c^(3) .

If |a| lt 1, |b| lt 1 , then show that a(a+b) + a^(2) (a^(2) + b^(2)) + a^(3)(a^(3) + b^(3)) +… = (a^(2))/(1-a^(2)) + (ab)/(1- ab)

By using properties of determinants , show that : (i) {:|( 1,a,a^(2)),( 1,b,b^(2)),( 1,c,c^(2))|:}=(a-b)(b-c) (c-a) (ii) {:|( 1,1,1),( a,b,c) ,(a^(3) , b^(3), c^(3))|:} =( a-b) (b-c)( c-a) (a+b+c)

Applying vectors , show that (a_(1)b_(1)+a_(2)b_(2)+a_(3)b_(3))^(2)le (a_(1)^(2)+a_(2)^(2)+a_(3)^(2))(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))

If a ,b ,c ,d are in G.P., then prove that (a^3+b^3)^(-1),(b^3+c^3)^(-1),(c^3+d^3)^(-1) are also in G.P.

a^(3)-b^(3)+1+3ab

If a=-1,b=2,c=3," then"(a^(3)+b^(3)+c^(3)-3abc)/((a-b)^(2)+(b-c)^(2)+(c-a)^(2))=

If x = a^((1)/(3)) b^((1)/(3)) + a^(-(1)/(3)) + a^(-(1)/(3)) b^((1)/(3)) then prove that a(bx^(3) - 3bx - a) = b^(2)

Examine whether the following sequences are in H.P. : {(1)/(a-3b), (1)/(a-2b), (1)/(a-b), (1)/(a),…}

If a = (sqrt5 + 1)/(sqrt5 + 1) and b = (sqrt5 -1)/(sqrt5 + 1) , then find the value of (a) (a^(2) + ab + b^(2))/(a^(2) - ab + b^(2)) (b) ((a -b)^(3))/((a + b)^(3)) (c) (3a^(2) + 5ab + b^(2))/(3a^(2) - 5ab + b^(2)) (d) (a^(3) + b^(3))/(a^(3) - b^(3))