Question

Complete the following series: $0,5,22,57,?,205.$

Answer 1

Complete the sequence.

In the given sequence,$0,5,22,57,?,205.$

$$\begin{gathered} \begin{array}{l}{1}^3-1=0\\ 2^3-3=5\\ 3^3-5=22\\ 4^3-7=57\end{array} \\ {6}^3-8=205 \end{gathered}$$

From the pattern above, it is observed as follows:

$\begin{array}{rcl}& & \begin{array}{c}{1}^{\mathrm{st}}\mathrm{term}:1^3-\left[\left(2\times 1\right)-1\right]=0\\ 2^{\mathrm{nd}}\mathrm{term}:2^3-\left[\left(2\times 2\right)-1\right]=5\\ 3^{\mathrm{rd}}\mathrm{term}:3^3-\left[\left(2\times 3\right)-1\right]=22\\ 4^{\mathrm{th}}\mathrm{term}:4^3-\left[\left(2\times 4\right)-1\right]=57\\ 6^{\mathrm{th}}\mathrm{term}:6^3-\left[\left(2\times 6\right)-1\right]=205\end{array}\end{array}$

So, the general form for $n\mathrm{th}$ term given sequence is $n^3-\left[\left(2\times n\right)-1\right]$

Since,$5^{th}$ the term is needed to find

Now, Put $n=5$ in the standard form $n^3-\left[\left(2\times n\right)-1\right]$:

$$\begin{gathered} n^3-\left[\left(2\times n\right)-1\right] \\ =5^3-\left[\left(2\times 5\right)-1\right] \\ =116 \end{gathered}$$

Hence the term is $116$

Hence, The complete series is $0,5,22,57,116,205.$