Question

If p^th, q^th and r^th terms of a G.P. are x, y, z respectively, find the value of x^{q – r} .y^{r – p} .z^{p – q}.

Answer 1

Let a be the first term and R be the common ratio of the G.P.
∴ t_n = a.R^n–1
∴ x = a.R^p–1, y = a.R^q–1, z = a.R^r–1
∴ x^q–r .y^r–p .z^p–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