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प्रश्न
Answer the following:
If p, q, r are in G.P. and `"p"^(1/x) = "q"^(1/y) = "r"^(1/z)`, verify whether x, y, z are in A.P. or G.P. or neither.
उत्तर
p, q, r are in G.P.
∴ `"q"/"p" = "r"/"q"`
∴ q2 = pr ...(1)
Also, `"p"^(1/x) = "q"^(1/y) = "r"^(1/z)`
∴ p = `"q"^(x/y)` and r = `"q"^(z/y)`
Substituting the values of p and r in (1), we get,
q2 = `"q"^(x/y)*"q"^(z/y)`
∴ q2 = `"q"^(x/y + z/y) = "q"^((x + z)/y)`
∴ 2 = `(x + z)/y`
∴ 2y = x + z
∴ y – x = z – y
Hence, x, y, z are in AP.
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