If (p + q)^th and (p − q)^th terms of a G.P. are m and n respectively, then write is pth term.

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Solution

$Here, {(p + q)}^{th} term = m \Rightarrow a r^{(p + q) - 1} = m . . . . . . . (i) And, {(p - q)}^{th} term = n \Rightarrow a r^{(p - q) - 1} = n . . . . . . . (ii) Dividing (i) by (ii) : \frac{a r^{(p + q) - 1}}{a r^{(p - q) - 1}} = \frac{m}{n} \Rightarrow r^{2 q} = \frac{m}{n} \Rightarrow r^{q} = \sqrt{\frac{m}{n}} Now, from (i) : a (r^{p - 1} \times r^{q}) = m \Rightarrow a r^{p - 1} \times \sqrt{\frac{m}{n}} = m \Rightarrow a r^{p - 1} = m \times \frac{\sqrt{n}}{\sqrt{m}} \Rightarrow a r^{p - 1} = \frac{m \sqrt{n}}{\sqrt{m}} Thus, the p^{t h} term is \frac{m \sqrt{n}}{\sqrt{m}} .$

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