If the `(p+q)^(th)` term of a G.P. is `a` and `(p-q)^(th)` term is `b`, determine its `p^(th)` term.

Explanation

` a_(p + q) = Ar^(p + q + 1 ) = a ` …(i)
` a_(p-q) = Ar^(p - q- 1) = b ` ….(ii)
Dividing (i) by (ii)
` r^( p + q - 1 - p + q + 1) = (a)/(b)`
`rArr r^(2q) = (a)/(b)`
and `A = (a)/(r^(p + q -1)) = (a)/(((a)/(b))^((p + q-1)/(2q)) )`
`therefore a_(p) = A r^(p-1) = (a)/(((a)/(b))^((p + q-1)/(2q)) )xx((a)/(b))^((p-1)/(2q))`
` (a) 1 + (p-1)/(2q) - (p + q -1)/(2q) (b)(p + q -1)/(2q) - (p-1)/(2q)`
` = (a)^((1)/(2)) (b)^((1)/(2)) = sqrt(ab)`
` p^(th) " term"= sqrt(ab)`
Alternative Method :
Multiply equation (i) and (ii) , ` Ar^(2p -2) = ab `
. or ` Ar^(p-1) = sqrt(ab)`