# For the following arithmetic sequence \begin{Bmatrix} -10,-2,6,14, ... \end{Bmatrix}, find a...

## Question:

For the following arithmetic sequence {eq}\begin{Bmatrix} -10,-2,6,14, ... \end{Bmatrix} {/eq}, find a recursive rule for the nth term.

## Arithmetic sequences:

Any sequence of real number is a function from set of natural numbers to the set of real numbers. Arithmetic sequences are special types of sequences in which difference between two successive terms are always constant and common.

Given:

• The given arithmetic sequence is {eq}\left\{ -10,-2,6,14,\ldots \right\}{/eq}.

Consider the given sequence:

The first term is {eq}{{a}_{1}}=-10{/eq} and the second term is {eq}{{a}_{2}}=-2{/eq}.

Since the given sequence is an arithmetic sequence, therefore there will be a common difference {eq}d{/eq} between any two successive terms, which can be calculated as:

{eq}\begin{align} d& ={{a}_{2}}-{{a}_{1}} \\ & =-2-\left( -10 \right) \\ & =8 \end{align}{/eq}

Since each successive terms has a fixed common difference with its previous terms, the {eq}{{\left( n+1 \right)}^{th}}{/eq} terms can be calculated by the addition of {eq}{{n}^{th}}{/eq} term with the common difference {eq}d{/eq}.

Hence the recursive formula for the given arithmetic sequence is:

{eq}{{a}_{n+1}}={{a}_{n}}+8{/eq} for {eq}n>1{/eq}, {eq}{{a}_{1}}=-10{/eq}.