Question

Justify if the series $\sqrt{3},\sqrt{12},\sqrt{27},\sqrt{48},...$ form an AP ?

Answer 1

Determine if the series form an AP

An AP is sequence of numbers such that the difference in each of consecutive terms are always same. The general form of the sequence is $a,a+d,a+2d,a+3d,....$ . Here the difference in each of consecutive term is $d$.

Check if the series form an AP:

The given series is in the form, $\sqrt{3},\sqrt{12},\sqrt{27},\sqrt{48},...$.

Let $a_n$ represents the nth term of the given sequence.

Then, $a_1=\sqrt{3},a_2=\sqrt{12},a_3=\sqrt{27},a_4=\sqrt{48}$.

The difference in first two consecutive term is $a_2-a_1=\sqrt{12}-\sqrt{3}=2\sqrt{3}-\sqrt{3}=\sqrt{3}$.

The difference in third and second term is $a_3-a_2=\sqrt{27}-\sqrt{12}=3\sqrt{3}-2\sqrt{3}=\sqrt{3}$.

The difference in forth and third term is $a_4-a_3=\sqrt{48}-\sqrt{27}=4\sqrt{3}-3\sqrt{3}=\sqrt{3}$.

Since, the difference in each of consecutive terms are same and is equal to $\sqrt{3}$.

Hence,The given sequence form an AP.