If a, b, c be in continued proportional, then prove that `a^2 b^2 c^2 (1/a^3 + 1/b^3+1/c^3) =a^3 +b^3+c^3`

If a,b,c, are in continued proportion then prove that a^2b^2c^2(1/a^3+1/b^3+1/c^3)= a^3+b^3+c^3 .

If a,b,c,d are in continued proportion,prove that: a:b+d=c^(3):c^(2)d+d^(3).

If a, b, c are in GP, prove that a^ 2 b^ 2 c ^ 2 ( 1/ a ^ 3 +1/ b ^ 3 ​ +1/ c ^ 3 ​ )=a ^ 3 +b^ 3 +c^ 3 .

(i) If a , b , c are in continued proportion, show that : (a^(2) + b^(2))/(b(a+c)) = (b(a + c))/(b^(2) + c^(2)) . (ii) If a , b , c are in continued proportion and a(b - c) = 2b , prove that : a - c = (2(a + b))/(a) . (iii) If (a)/(b) = (c)/(d) show that : (a^(3)c + ac^(3))/(b^(3)d +bd^(3)) = ((a + c)^(4))/((b + d)^(4)) .

Prove that (a^8+b^8+c^8)/(a^3b^3c^3)>1/a+1/b+1/c

Prove that (a^8+b^8+c^8)/(a^3b^3c^3)>1/a+1/b+1/c

If a + b + c = 0 , then prove that (a^2+b^2+c^2)/(a^3+b^3+c^3)+2/3(1/a+1/b+1/c)=0

If 2s = a + b + c , Then prove that (s-a)^3+(s-b)^3+(s-c)^3-3(s-a)(s-b)(s-c) = 1/2(a^3+b^3+c^3-3abc)

Prove that a^3+b^3+c^3-3abc=1/2(a+b+c){(a-b)^2+(b-c)^2+(c-a)^2}

Without expanding the determinant, prove that |(a,a^2,bc),(b,b^2,ca),(c,c^2,ab)|=|(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3)|