Question

Find the next term of the AP: $3,1,-1,-3,...$

Answer 1

Calculate the next term of the arithmetic progression:

Given arithmetic progression, $3,1,-1,-3,...$.

The first term of the arithmetic progression, $a_1=3$.

As the given series is an arithmetic progression, thus two consecutive terms must have a common difference.

The common difference of the arithmetic progression,

$\begin{array}{rcl}d& =& a_2-a_1\\ & =& 1-3\\ & =& -2\end{array}$.

The $n^{th}$ term of the progression is given as,

${\mathit{a}}_{\mathbf{n}}\mathbf{=}{\mathit{a}}_{\mathbf{1}}\mathbf{+}\left(n-1\right)\mathit{d}$

$$\begin{gathered} \Rightarrow a_{\mathrm{n}}=3+\left(n-1\right)\left(-2\right) \\ \Rightarrow a_{\mathrm{n}}=3-2\left(n-1\right) \end{gathered}$$

Thus, the next term of the series can be given by,

$a_5=3-2\left(5-1\right)$.

$$\begin{gathered} \Rightarrow a_5=3-\left(2\times 4\right) \\ \Rightarrow a_5=3-8 \\ \Rightarrow a_5=-5 \end{gathered}$$

Therefore, the next term of the given arithmetic progression is $\mathrm{-}\mathrm{5}$.