Q : 4         Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.

                  (xii)    \small \sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32},...

Answers (1)

Given series is
\small \sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32},...
We can rewrite it as 
\sqrt2,2\sqrt2,3\sqrt2,4\sqrt2,....
Now,
first term to this series is = a
Now,
a_1 = \sqrt2 \ \ and \ \ a_2 = 2\sqrt2 \ \ and \ \ a_3 = 3\sqrt2 \ \ and \ \ a_4 = 4\sqrt2
a_2-a_1 = 2\sqrt2-\sqrt2 =\sqrt2
a_3-a_2 = 3\sqrt2-2\sqrt2 =\sqrt2
a_4-a_3=4\sqrt2-3\sqrt2=\sqrt2
We can clearly see that difference between terms are equal  and equal to \sqrt2
Hence, given series is in AP
Now, next three terms are
a_5=a_4+d =4\sqrt2+\sqrt2=5\sqrt2
a_6=a_5+d =5\sqrt2+\sqrt2=6\sqrt2
a_7=a_6+d =6\sqrt2+\sqrt2=7\sqrt2
Therefore, next three terms of given series are 5\sqrt2,6\sqrt2,7\sqrt2

That is the next three terms are \sqrt{50},\ \sqrt{72},\ \sqrt{98}

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