Question

What is the next term in the sequence: `sqrtx/3, (2sqrtx)/3, sqrtx,...`?

Answer 1

`(4sqrt(x))/3`

Explanation:

Given:

`sqrt(x)/3, (2sqrt(x))/3, sqrt(x)`

We could also write this as:

`(1sqrt(x))/3, (2sqrt(x))/3, (3sqrt(x))/3`

This is an arithmetic sequence, with common difference `sqrt(x)/3`

So (if it continues as an arithmetic sequence) the next term is formed by adding the common difference.

`(3sqrt(x))/3 + sqrt(x)/3 = (4sqrt(x))/3`

Answer 2

Next term, I.e. the `4^(th)` term `a_4 = color(blue)((4sqrtx)/3)`

Explanation:

Difference between `2^(nd)` & `1^(st)` term is

`(2sqrtx)/3 - sqrtx/3 = sqrtx/3`

Similarly, difference between `3^(rd)` & `2^(nd)` term is `(sqrtx/3)`

Therefore, the common difference between successive terms `d = sqrtx/3`

First term `a = sqrtx/3`

This is an arithmetic progression (A.P) with `a = sqrtx/3, d = sqrtx/3`

`n^th` term of A.P is given by the formula

Sa_n = a + ((n-1)*d) #

Fourth term `a_4 = (sqrtx/3) + ((4-1)*sqrtx/3)`

`a_4 = (sqrtx/3 + (3sqrtx)/3) = (4sqrtx)/3`