Question

If the first and the n th term of a G.P. are a ad b , respectively, and if P is the product of n terms, prove that P 2 = ( ab ) n .

Answer 1

Given that the first term and n th term of G.P are a and b respectively and P is the product of n terms.

Let the first term of the G.P. is a and the last term is b and the G.P is a, ar, a r 2 ,........a r n−1 .

Now, the last term of the G.P is,

b=a r n−1

Now, given that P is product of n terms, then,

P=Product of n terms =a×( ar )×( a r 2 )×.........×( a r n−1 ) =( a×a×......×a )×( r× r 2 ×.......× r n−1 ) = a n r 1+2+......+( n−1 )

Now, the power of common ratio is A.P.

1+2+...+( n−1 )

Now, the sum of n term of A.P is,

1+2+...+( n−1 )= ( n−1 ) 2 [ 2+( n−1−1 )×1 ] = ( n−1 ) 2 [ 2+n−2 ] = n( n−1 ) 2

The value of P is,

P= a n r n( n−1 ) 2

Squaring both side of above equation, we get

P 2 = ( a n r n( n−1 ) 2 ) 2 = a 2n r n( n−1 ) = [ a 2 r n−1 ] n = [ a×a r n−1 ] n

By using equation (1), we proved that

P 2 = ( ab ) n

Hence, it is proved.