Question

Find the next term in the series $2,3,8,63$.

Answer 1

Calculate the next term of the given series:

Given series, $2,3,8,63$.

It can be observed that the given series follows a certain pattern.

$2$ is the first term.

Then, the corresponding terms can be determined as below.

$$\begin{gathered} 3=4-1=2^2-1 \\ 8=9-1=3^2-1 \\ 63=64-1=8^2-1 \end{gathered}$$

So, the terms of the series can be determined by, subtracting $1$ from the square of the preceding number.

Thus, the $n^{th}$ term of the series can be given by, $t_n={\left(t_{n-1}\right)}^2-1$.

Hence, the next term of the series is,

$\begin{array}{rcl}{t}_5& =& {\left(t_{5-1}\right)}^2-1\\ & =& {\left(t_4\right)}^2-1\\ & =& {63}^2-1\\ & =& 3969-1\\ & =& 3968\end{array}$.

Therefore, the next term of the given series is $3968$.