Question

Do the irrational number root2 , root8, root18 , root32...form an A.P ? If so find the common difference.

Answer 1

Given sequence : √2 , √8 , √18 , √32 .....

⇒ √2 , √( 2 x 2 x 2 ) , √( 2 x 3 x 3 ) , √( 2 x 2 x 2 x 2 x 2 ).....

⇒ √2 , √( 2 x 2^2 ) , √( 2 x 3^2 ) , √( 2 x 4^2 ).....

⇒ √2 , 2√2 , 3√2 , 4√2 .....

Now, 1st term = √2

        2nd term = 2√2

        3rd term = 3√2

        4th term = 4√2

We know that in arithmetic progressions difference between the consecutive terms cannot be changed.

So, difference of any two consecutive terms will be equal to the difference between other two consecutive terms.

Now, checking that the given sequence is a AS or not by taking out the difference between them.

Difference between 2nd and 1st term = 2√2 - √2 = √2

Difference between 3rd and 2nd term = 3√2 - 2√2 = √2

Difference between 4th and 3rd term = 4√2 - 3√2 = √2

As √2 is the difference which is common in all consecutive terms of the given series, given irrational number form an AP( arithmetic progressions ).