# 17.  If the $4 ^{th} , 10 ^{th} , 16 ^ {th}$  terms of a G.P. are x, y and z, respectively. Prove that x,y, z are in G.P.

Let x,y, z are in G.P.

Let first term=a and common ratio = r

$a_4=a.r^3=x..................(1)$

$a_1_0=a.r^9=y..................(2)$

$a_1_6=a.r^1^5=z..................(3)$

Dividing equation 2 by 1, we have

$\frac{a.r^9}{a.r^3}=\frac{y}{x}$

$\Rightarrow r^4=\frac{y}{x}$

Dividing equation 3 by 2, we have

$\frac{a.r^1^5}{a.r^9}=\frac{z}{y}$

$\Rightarrow r^4=\frac{z}{y}$

Equating values of $r^4$ ,  we have

$\frac{y}{x}=\frac{z}{y}$

Thus, x,y,z are in GP

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