16. If a, b, c are in GP. And a^x = b^y=c^z=then prove that 1/x+1/z =2/y.
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Answer:
a^x = b^x = c²
a, b, c→ GP
b² = ac
xloga = ylogb = zlogc
xloga = y/2 logc
xloga = y/2 (loga + loge) = 2
(2x - y)loga = ylogc
loga/logc = y/2x-y (1)
y/2 logac = zlogc
y/2 (loga+logc) = zlogc
yloga = (2z - y)logc
loga/logc = 2z-y/y (2)
(1) = (2)
y/2x-y = 2z-y/y
y² = (2x - y) (2z - y)
4 x z-y (2x + 2z) = 0
2 x 2 − y(x+2) = 0
2/y = 1/z + 1/x
Step-by-step explanation:
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