Question

If x + z = 2y and b2 = ac, then let us show that a^y-z b^z-x c^x-y=1​

Answer 1

Answer:

Since x + z = 2y, z = 2y - x and:

b^(z - x) = b^(2y - 2x) = (b^2)^(y - x)

But b^2 = ac, so that becomes:

b^(z - x) = a^(y - x) c^(y - x)

a^(y - z) b^(z - x) c^(x - y) = a^(y - z + y - x) c^(x - y + y - x)

= a^(2y - z - x) c^0

= a^(2y - z - x)

But x + z = 2y also means 2y - z - x = 0, so the exponent on a is zero too

a^(y - z) b^(z - x) c^(x - y) = a^(0) = 1

This, of course, requires a,b,c to be nonzero.