If a,b,c are in geometric progression and if `a^((1)/(x))=b^((1)/(y))=c^((1)/(z))`, then prove that x,y,z are in arithmetic progression.

If a,b,c are in geometric progression, and if a^(1/x) = b^(1/y) =c^(1/z) , then prove that x,y,z arithmetic progression.

If a,b,c are in G.P and a^(1/x) = b^(1/y) = c^(1/z) , prove that x,y,z are in A.P.

If the roots of (a-b)x^(2)+(b-c)x+(c-a)=0 are real and equal, then prove that b, a, c are in arithmetic progression.

The first term of an arithmetic progression is 1 and the sum of the first nine terms equal to 369 . The first and the ninth term of a geometric progression coincide with the first and the ninth term of the arithmetic progression. Find the seventh term of the geometric progression.

If y^2=x z and a^x=b^y=c^z , then prove that (log)_ab=(log)_bc

If a , b ,c are in harmonic progression, then the straight line ((x/a))+(y/b)+(1/c)=0 always passes through a fixed point. Find that point.

The value of x + y + z is 15. If a, x, y, z, b are in AP while the value of (1)/(x) + (1)/(y) + (1)/(z) " is " (5)/(3) . If a, x, y, z b are in HP, then find a and b .

If ("log"x)/(y - z) = ("log" y)/(z - x) = ("log" z)/(x - y) , then prove that xyz = 1.

Write the first 6 terms of the sequences whose n^(th) terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them. 1/2^(n+1)