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Question
Answer the following:
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.
Then tn = ARn–1
Now, tp = x, tq = y and tr = z
∴ ARp–1 = x, ARq–1 = y and ARr–1 = z
∴ xq–r .yr–p .zp–q
= (ARp–1)q–r . (ARq–1)r–p . (ARr–1)p–q
`="A"^("q"–"r") * "R"^("pq"–"pr"–"q"+"r") * "A"^("r"–"p") * "R"^("qr"–"pq"-"r"+"p") * "A"^("p"–"q") *"R"^("pr"–"qr"–"p"+"q")`
= `"A"^("q"–"r"+"r"–"p"+"p"-"q") * "R"^("pq"–"pr"–"q"+"r"+"qr"–"pq"–"r"+"p"+"pr"–"qr"–""p"+"q")`
= A° · R°
= 1
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