If the first and the n^th terms of a G.P are a and b respectively and if p is the product of the first n terms, prove that P² = (ab)ⁿ.

Let the G.P be c, cr, cr², ……….crⁿ

1^st term c = a

n^th term cr ^n-1 = b

If n terms is c, cr, cr², ……….cr^n-1

Product of n term

p = c.(cr)(cr²)(cr³)……..(cr^n-1) 

   = cⁿ.r^{1+2+………n-1}

   = cⁿ 

P² = (cⁿ)^{2 2} = c²ⁿr^n(n-1) = c²ⁿ _rn^2-n ………….(1)

_(ab)n _{= (c.cr}n-1)n _{= (c}2_rn-1₎n _{= c}2n _rn^2-n ……….(2)

From (1) and (2),

P² = (ab)ⁿ.