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Question
If pth, qth and rth terms of a G.P. are x, y, z respectively, find the value of xq – r .yr – p .zp – q.
Solution
Let a be the first term and R be the common ratio of the G.P.
∴ tn = a.Rn–1
∴ x = a.Rp–1, y = a.Rq–1, z = a.Rr–1
∴ xq–r .yr–p .zp–q
= `("a.R"^("p"–1))^("q–r") .("a.R"^("q"–1))^("r–p") .("a.R"^("r"–1))^("p–q")`
= `"a"^("q–r")"R"^(("p"–1)("q"–r))*"a"^("r–p")"R"^(("q"–1)("r"–p))*"a"^("p–q")"R"^(("r"–1)("p–q"))`
= `"a"^(("q" - "r" + "r" - "p" + "p" - "q"))*"R"^([("p" - 1)("q - r") + ("q" - 1)("r - p") + ("r" - 1)("p - q")])`
= `"a"^0*"R"^(("pq - pr - q + r + qr + - pq - r + p + pr - qr - p + q")`
= (1).R0
= 1
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