The start of an arithmetic sequence is shown below. What is the n^th term rule for this sequence? The n^th term rule is square n-square

## Step1: Determine the common difference<br />Look at the sequence \(3, 12, 21 , 30 , \ldots\). We have two ways to generate 12 using 3, either multiply it by 4 or add 9 to it. Similarly for 12 and 21, either multiply 12 by 1.75 or add 9 to it, and so on. Clearly, adding 9 is the constant operation across all transitions from one term to the next. This 9 is the "common difference" in an arithmetic sequence. The rule for an arithmetic sequence is typically expressed as \(a_n = dn + c \) or \( a_n = d(n-1) + a_1 \), where \( d \) is the common difference, \( a_1 \) is the first term, and \( c \) is a constant.<br /><br />### Formulas:<br />1. Difference: \(d = a_2 - a_1\) (where \(a_2\) and \(a_1\) are the second and first term respectively)<br />2. Arithmetic Cquence Rule: \(a_n = a_1 + d(n - 1)\) <br /><br />## Step2: Subtitute the values from our sequence into the formula<br />Substituting the value we find that \(a_1 = 3\) and \(d = 9\), into the Arithmetic Sequence Rule formula we get:<br /><br />### Formula After Substitution:<br /> \(a_n = 3 + 9(n - 1) = 9n- 6\)