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Question
If p − q is small compared to either p or q, then show `root("n")("p"/"q")` ∼ `(("n" + 1)"p" + ("n" - 1)"q")/(("n"- 1)"p" +("n" + 1)"q")`. Hence find `root(8)(15/16)`
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Solution
R.H.S = `(("n" + 1)"p" + ("n" - 1)"q")/(("n"- 1)"p" + ("n" + 1)"q")`
= `("n"("p" + "q") + ("p" - "q"))/("n"("p" + "q") - ("p" - "q)`
=`(1 + 1/"n" (("p" - "q")/("p" + "q")))/(1 - 1/"n"(("p" - "q")/("p" + "q"))`
= `(1 + ("p"- "q")/("p" + "q"))^(1/"n")/(1 - ("p"- "q")/("p" + "q"))^(1/"n")`
= `("p"/4)^(1/"n")`
= `root("n")("p"/"q")`
= L.H.S
To find `root(8)(15/16)`
We take n = 8, p = 15, q = 16
So `root(8)(15/16)`
= `(("n" + 1)"p" + ("n" - 1)"q")/(("n" - 1)"p" + ("n" + 1)"q")`
= `(9 xx 15 + 7 xx 16)/(7 xx 15 + 9xx 16)`
= `(135 + 112)/(105 + 144)`
= `247/249`
= 0.99196
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