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Question
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
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Solution
p, q, r, s are in G.P.
∴ `"q"/"p" = "r"/"q" = "s"/"r"`
Let `"q"/"p" = "r"/"q" = "s"/"r"` = k
∴ q = pk, r = qk, s = k
We have to prove that p + q, q + r, r + s are in G.P.
i.e. to prove that `("q" + "r")/("p" + "q") = ("r" + "s")/("q" + "r")`
L.H.S. = `("q" + "r")/("p" + "q") = ("q" + "qk")/("p" + "pk") = ("q"(1 + "k"))/("p"(1 + "k")) = "q"/"p"` = k
R.H.S. = `("r" + "s")/("q" + "r") = ("r" + "rk")/("q" + "qk") = ("r"(1 + "k"))/("q"(1 + "k")) = "r"/"q"` = k
∴ `("q" + "r")/("p" + "q") = ("r" + "s")/("q" + "r")`
∴ p + q, q + r, r + s are in G.P.
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