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Progressions — Maths STD 10 — Question

Gujarat BoardEnglish MediumSTD 10MathsProgressions2 Marks
Question
Is this $3,3 + \sqrt 2 ,3 + 2\sqrt 2 ,3 + 3\sqrt 2$, ..... an $AP$? If it forms an $AP,$ find the common difference $d$ and write three more terms.
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Answer

$3,3 + \sqrt 2 ,3 + 2\sqrt 2 ,3 + 3\sqrt 2 ,....$
${a_2} - {a_1} = (3 + \sqrt 2 ) - 3 = \sqrt 2 $
${a_3} - {a_2} = (3 + 2\sqrt 2 ) - (3 + \sqrt 2 ) = \sqrt 2 $
${a_4} - {a_3} = (3 + 3\sqrt 2 ) - (3 + 2\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 differenced d = $\sqrt 2 $
The next three terms are:
$(3 + 3\sqrt 2 ) + \sqrt 2 = 3 + 4\sqrt 2 ,$ $(3 + 4\sqrt 2 ) + \sqrt 2 = 3 + 5\sqrt 2 $ and $(3 + 5\sqrt 2 ) + \sqrt 2 = 3 + 6\sqrt 2 $

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