The table below gives the positions of three terms in an arithmetic sequence What is the n^ (th ) term rule for the sequence? Position, n & ldots & 100 & 101 & 102 Term & ldots & 324 & 327 & 330

## Step 1: Identify the common difference of the arithmetic sequence<br />By looking at the given numbers in the sequence, namely \(324\), \(327\), and \(330\), we can see that the common difference, \(d\), is \(+3\).<br /><br />## Step 2: Identify the first term of the sequence<br />Since the common difference is known, the first term of the sequence can be found by subtracting [(position - 1) \(\cdot\) common difference] from any known term. Let's take the \(100^{th}\) term, subtracting \(99 \cdot 3\) (This consists of 99 steps of +3 from the first term)<br /><br />### \(a _ { 1 } = a - (n-1) \cdot d\)<br />### \(a_{1} = 324 - (100-1) \cdot 3\)<br />### \(a_{1} = 324 - 297 = 27\)<br /><br />## Step 3: Write out the nth term rule for the sequence<br />The "nth term" or general rule of an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(d\) is the common difference and \(n\) is the term number. Substituting the values that we computed before for \(a_1\) and \(d\), the formula becomes<br /><br />### \(a_n = 27 + 3(n-1)\)<br />This formula allows you to find any term in the sequence knowing its position.