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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.
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 = rk
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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