Question
Is this $\sqrt 2 ,\sqrt 8 ,\sqrt {18} ,\sqrt {32} ,...$ an $AP?$ If it forms an $AP$, find the common difference $d$ and write three more terms.
✓
Answer
$\sqrt 2 ,\sqrt 8 ,\sqrt {18} ,\sqrt {32} ,...$
${a_2} - {a_1} = \sqrt 8 - \sqrt 2 = 2\sqrt 2 - \sqrt 2 = \sqrt 2 $
${a_3} - {a_2} = \sqrt {18} - \sqrt 8 = 3\sqrt 2 - 2\sqrt 2 = \sqrt 2 $
${a_4} - {a_3} = \sqrt {32} - \sqrt {18} = 4\sqrt 2 - 3\sqrt 2 = \sqrt 2 $
i.e. $a_{k+1}- a_k$ is the same every time.
So, the given list of numbers forms an $AP$ with the common difference d = $\sqrt 2 .$
The next three terms are:
$\sqrt {32} + \sqrt 2 = 4\sqrt 2 + \sqrt 2 = 5\sqrt 2 = \sqrt {50} $
$5\sqrt 2 + \sqrt 2 = 6\sqrt 2 = \sqrt {72} $
and $6\sqrt 2 + \sqrt 2 = 7\sqrt 2 = \sqrt {98} $
${a_2} - {a_1} = \sqrt 8 - \sqrt 2 = 2\sqrt 2 - \sqrt 2 = \sqrt 2 $
${a_3} - {a_2} = \sqrt {18} - \sqrt 8 = 3\sqrt 2 - 2\sqrt 2 = \sqrt 2 $
${a_4} - {a_3} = \sqrt {32} - \sqrt {18} = 4\sqrt 2 - 3\sqrt 2 = \sqrt 2 $
i.e. $a_{k+1}- a_k$ is the same every time.
So, the given list of numbers forms an $AP$ with the common difference d = $\sqrt 2 .$
The next three terms are:
$\sqrt {32} + \sqrt 2 = 4\sqrt 2 + \sqrt 2 = 5\sqrt 2 = \sqrt {50} $
$5\sqrt 2 + \sqrt 2 = 6\sqrt 2 = \sqrt {72} $
and $6\sqrt 2 + \sqrt 2 = 7\sqrt 2 = \sqrt {98} $
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