Verify that each of the following is an $ \mathrm{AP} $, and then write its next three terms.
$ \sqrt{3}, 2 \sqrt{3}, 3 \sqrt{3}, \ldots $
Given:
Given sequence is \( \sqrt{3}, 2 \sqrt{3}, 3 \sqrt{3}, \ldots \)
To do:
We have to verify whether the given sequence is an AP and write its next three terms.
Solution:
In the given sequence,
$a_1=\sqrt{3}, a_2= 2\sqrt{3}, a_3=3\sqrt{3}$
$a_2-a_1=2\sqrt{3}-\sqrt{3}=\sqrt{3}$
$a_3-a_2=3\sqrt{3}-2\sqrt{3}=\sqrt{3}$
Therefore,
$a_2-a_1=a_3-a_2$
The given sequence is an AP.
$d=\sqrt{3}$
$a_4=a_3+d=3\sqrt{3}+\sqrt{3}=4\sqrt{3}$
$a_5=a_4+d=4\sqrt{3}+\sqrt{3}=5\sqrt{3}$
$a_6=a_5+d=5\sqrt{3}+\sqrt{3}=6\sqrt{3}$
The next three terms of the given sequence are $4\sqrt3, 5\sqrt3$ and $6\sqrt3$.
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