(i) The given arrangement,
![\[2,\text{ }2\surd 2,\text{ }4,\ldots \]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ce00ccd652a261071920c1a1d866c46e_l3.png "Rendered by QuickLaTeX.com")
We have,
![\[a\text{ }=\text{ }2\text{ }and\text{ }r\text{ }=\text{ }2\surd 2/2\text{ }=\text{ }\surd 2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ef09703224e22e346634c7cf2bbc6b39_l3.png "Rendered by QuickLaTeX.com")
Taking the
![\[{{n}^{th}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d980edc5c4f51eb25f01ef4772027813_l3.png "Rendered by QuickLaTeX.com")
term of this arrangement as
![\[128\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0d29c433b0b664c4733be0be24e2e1fe_l3.png "Rendered by QuickLaTeX.com")
, we have
Accordingly, the thirteenth term of the given succession is
![\[128.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-303b5858329805ea1ae043fa8c1d5791_l3.png "Rendered by QuickLaTeX.com")
(ii) Given arrangement,
![\[\surd 3,\text{ }3,\text{ }3\surd 3,\ldots \]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b36746a5eb50b8502320f74de5b4c822_l3.png "Rendered by QuickLaTeX.com")
We have,
![\[a\text{ }=\text{ }\surd 3\text{ }and\text{ }r\text{ }=\text{ }3/\surd 3\text{ }=\text{ }\surd 3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-25c533af08e8703004fcb21b24de617c_l3.png "Rendered by QuickLaTeX.com")
Taking the nth term of this arrangement to be
![\[729,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f5c94da868dd94713d5b41536cee0282_l3.png "Rendered by QuickLaTeX.com")
we have
Accordingly, the twelfth term of the given succession is
![\[729.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ff6f2f266cf4d87303d7e87d8cb9ab48_l3.png "Rendered by QuickLaTeX.com")