Question

Find the next number in the sequence $6,24,60,120,...$

Answer 1

Calculate the next term of the given sequence:

Given sequence, $6,24,60,120,...$.

The given sequence is following a certain pattern.

$$\begin{gathered} 6=8-2=2^3-2 \\ 24=27-3=3^3-3 \\ 60=64-4=4^3-4 \\ 120=125-5=5^3-5 \end{gathered}$$

Thus, the $n^{th}$ term of the sequence can be given by, $t_n={\left(n+1\right)}^3-\left(n+1\right)$.

So, the next term of the sequence can be determined as,

$\begin{array}{rcl}{t}_5& =& {\left(5+1\right)}^3-\left(5+1\right)\\ & =& {\left(6\right)}^3-6\\ & =& 216-6\\ & =& 210\end{array}$.

Therefore, the next term of the given sequence is $210$.