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23.  If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that  

        P^2 = ( ab)^n.

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Given : First term =a  and n th term = b.

Common ratio = r.

To prove : P^2 = ( ab)^n

Then , GP = a,ar,ar^2,ar^3,ar^4,..........................

a_n=a.r^{n-1}=b..................................1

P = product of n terms

P=(a).(ar).(ar^2).(ar^3)..............(ar^{n-1})

P=(a.a.a...............a)((1).(r).(r^2).(r^3)..............(r^{n-1}))

P=(a^n)(r^{1+2+.........(n-1)})........................................2

Here, 1+2+.........(n-1)   is a AP.

\therefore\, \, \, sum= \frac{n}{2}\left [2a+(n-1)d \right ]

                   = \frac{n-1}{2}\left [2(1)+(n-1-1)1 \right ]

                   = \frac{n-1}{2}\left [2+n-2 \right ]

                    = \frac{n-1}{2}\left [n \right ]

                   = \frac{n(n-1)}{2}

Put in equation (2),

P=(a^n)(r^{\frac{n(n-1)}{2}})

P^2=(a^2^n)(r^{n(n-1)})

P^2=(a. a.r^{(n-1)})^n

P^2=(a.b)^n

 Hence proved .

Posted by

seema garhwal

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