Login/Sign Up
Home » If the fourth, seventh and tenth terms of a G.P. are x, y, z respectively, prove that x, y, z are in G.P.
If the fourth, seventh and tenth terms of a G.P. are x, y, z respectively, prove that x, y, z are in G.P.
Arithmetic and Geometric Progression | Class 10 | Exercise 1 | ICSE | Maths | ML Aggarwal
If the fourth, seventh and tenth terms of a G.P. are x, y, z respectively, prove that x, y, z are in G.P.

From the question it is given that,

a4 = x

a7 = y

a10 = z

 

Now we have to prove that, x, y, z are in G.P.

Then, by the formula

    \[{{a}_{n}}~=\text{ }a{{r}^{n\text{ }\text{ }1}}\]

    \[\begin{array}{*{35}{l}} {{a}_{4}}~=\text{ }a{{r}^{4\text{ }\text{ }1}} \\ {{a}_{4}}~=\text{ }{{a}^{3}} \\ {{a}_{4}}~=\text{ }x \\ \end{array}\]

So,

    \[{{a}_{7}}~=\text{ }{{a}^{7\text{ }\text{ }1}}\]

    \[\begin{array}{*{35}{l}} {{a}_{7}}~=\text{ }{{a}^{6}} \\ {{a}_{7}}~=\text{ }y \\ {{a}_{10}}~=\text{ }{{a}^{10\text{ }-1}} \\ {{a}_{10}}~=\text{ }{{a}^{9}} \\ {{a}_{10}}~=\text{ }z \\ \end{array}\]

x, y, z are in G.P. then,

y2 = xz

Then, xz =

    \[a{{r}^{3}}~\times \text{ }a{{r}^{9}}\]

    \[\begin{array}{*{35}{l}} =\text{ }{{a}^{2}}{{r}^{3}}{{~}^{+\text{ }9}} \\ =\text{ }{{a}^{2}}{{r}^{12}} \\ {{y}^{2}}~=\text{ }{{(a{{r}^{6}})}^{2}} \\ {{y}^{2}}~=\text{ }{{a}^{2}}{{r}^{12}} \\ \end{array}\]

By comparing left hand side and right hand side

LHS = RHS

Therefore, x, y and z are in G.P.

i

More Material