Question

What is the next number in the series $0,6,24,60,120,210,$?

Answer 1

Step 1. Difference between the first and second terms in the series.

The given series is $0,6,24,60,120,210,$.

Take the difference between the first and second terms:

$\begin{array}{rcl}6-0& =& 6\\ 24-6& =& 18\\ 60-24& =& 36\\ 120-60& =& 60\\ 210-120& =& 90\end{array}$

Thus, the new series of differences is $6,18,36,60,90$.

Step 2: Difference between the first and second terms of the new series$6,18,36,60,90$.

Consider the sixth term of the series is $X$.

$6,18,36,60,90,X$

$\begin{array}{rcl}18-6& =& 12\\ 36-18& =& 18\\ 60-36& =& 24\\ 90-60& =& 30\end{array}$

Hence, $X-90=Y$

Thus, the new series of differences is $12,18,24,30,Y...\left(ii\right)$.

Step 3: Difference between the first and second terms of the new series$12,18,24,30,Y$.

Where the fifth term of the series is $Y$.

$12,18,24,30,Y$

$\begin{array}{rcl}18-12& =& 6\\ 24-18& =& 6\\ 30-24& =& 6\end{array}$

Hence, $Y-30=6$

Thus, the difference is $6$, which is constant.

Step 4. Find the next term of the given series.

Add $6$ to the fourth term of the series of $12,18,24,30$ in Step $2$ to find the value of $Y$:

$\begin{array}{rcl}Y& =& 30+6\\ & \Rightarrow & Y=36\end{array}$

Add $36$ to the fifth term of the series $6,18,36,60,90$ of Step $1$ to find the value of $X$:

$\begin{array}{rcl}X& =& 90+36\\ & \Rightarrow & X=126\end{array}$

Now, add $126$ to the sixth term of the given series:

$210+126=336$

Hence, the next term in the given series is $336$ and the series is $0,6,24,60,120,210,360$