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 differenceLook at the sequence . 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 or \( a_n = d(n-1) + a_1 \), where is the common difference, is the first term, and is a constant.### Formulas:1. Difference: (where and are the second and first term respectively)2. Arithmetic Cquence Rule: \(a_n = a_1 + d(n - 1)\) ## Step2: Subtitute the values from our sequence into the formulaSubstituting the value we find that and , into the Arithmetic Sequence Rule formula we get:### Formula After Substitution: \(a_n = 3 + 9(n - 1) = 9n- 6\)