Question

If a, b, c are three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P. Then prove that

Answer 1

$x^{b-c}. y^{c-a}.z^{a-b}$  = 1  proved  

Step-by-step explanation:

Given as :

a , b, c are three consecutive terms of an A.P

And

x , y , z  are three consecutive terms of a G.P

To prove : $x^{b-c}. y^{c-a}.z^{a-b}$ = 1

According to question

As a , b, c are three consecutive terms of an A.P

∴, b - a  = c - b  = d

where  a , b , c consecutive terms and d is common difference

∵ x , y , z  are three consecutive terms of a G.P

∴,  y  = $\sqrt{xz}$

from Left hand side

$x^{b-c}. y^{c-a}.z^{a-b}$  = $x^{-d}$  . $y^{2d}$ . $z^{-d}$

                         = $x^{-d}$ . $(\sqrt{xz} )^{2d}$ . $z^{-d}$                   ( ∵  y  = $\sqrt{xz}$ , from G.P )

                         = $x^{-d}$  . $x^{\dfrac{2d}{2}}$ . $z^{\dfrac{2d}{2}}$ . $z^{-d}$

                         =  $x^{-d}$ . $x^{d}$ . $z^{d}$ . $z^{-d}$

                         = $x^{-d+d}$ . $z^{d-d}$

                        = $x^{0}$ . $z^{0}$

                        = 1                                                 ( ∵ $a^{0}$  = 1 )

So, Left hand side of equation = Right hand side

Hence, $x^{b-c}. y^{c-a}.z^{a-b}$  = 1  proved  Answer

Answer 2

Step-by-step explanation:

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