Question

Write the first four terms in the list of numbers defined by $a_n=\frac{{\left(n-1\right)}^2}{3}$

Answer 1

Find the first four terms of sequence $a_n=\frac{{\left(n-1\right)}^2}{3}$ by substituting $n=1,2,3,4$:
$\begin{array}{rcl}& \Rightarrow & a_1=\frac{{\left(1-1\right)}^2}{3}\\ & =& 0\\ & \Rightarrow & a_2=\frac{{\left(2-1\right)}^2}{3}\\ & =& \frac{1}{3}\\ & \Rightarrow & a_3=\frac{{\left(3-1\right)}^2}{3}\\ & =& \frac{4}{3}\\ & \Rightarrow & a_4=\frac{{\left(4-1\right)}^2}{3}\\ & =& \frac{9}{3}\\ & =& 3\end{array}$

Therefore, the first four terms of sequence is $0,\frac{1}{3},\frac{4}{3},3$.