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

Oct 24, 2016

$\frac{4 \sqrt{x}}{3}$

#### Explanation:

Given:

$\frac{\sqrt{x}}{3} , \frac{2 \sqrt{x}}{3} , \sqrt{x}$

We could also write this as:

$\frac{1 \sqrt{x}}{3} , \frac{2 \sqrt{x}}{3} , \frac{3 \sqrt{x}}{3}$

This is an arithmetic sequence, with common difference $\frac{\sqrt{x}}{3}$

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

$\frac{3 \sqrt{x}}{3} + \frac{\sqrt{x}}{3} = \frac{4 \sqrt{x}}{3}$

Jan 18, 2018

Next term, I.e. the ${4}^{t h}$ term ${a}_{4} = \textcolor{b l u e}{\frac{4 \sqrt{x}}{3}}$

#### Explanation:

Difference between ${2}^{n d}$ & ${1}^{s t}$ term is

$\frac{2 \sqrt{x}}{3} - \frac{\sqrt{x}}{3} = \frac{\sqrt{x}}{3}$

Similarly, difference between ${3}^{r d}$ & ${2}^{n d}$ term is $\left(\frac{\sqrt{x}}{3}\right)$

Therefore, the common difference between successive terms $d = \frac{\sqrt{x}}{3}$

First term $a = \frac{\sqrt{x}}{3}$

This is an arithmetic progression (A.P) with $a = \frac{\sqrt{x}}{3} , d = \frac{\sqrt{x}}{3}$

${n}^{t} h$ term of A.P is given by the formula

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

Fourth term ${a}_{4} = \left(\frac{\sqrt{x}}{3}\right) + \left(\left(4 - 1\right) \cdot \frac{\sqrt{x}}{3}\right)$

${a}_{4} = \left(\frac{\sqrt{x}}{3} + \frac{3 \sqrt{x}}{3}\right) = \frac{4 \sqrt{x}}{3}$