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In arithmetic sequences, identifying the common difference is the initial and crucial step. The common difference is the amount by which each consecutive term increases or decreases in the sequence. To find it, subtract the first term from the second term. For instance, in the sequence given, where the terms are 12, 20, 28, and 36, you calculate the common difference as follows:

• Second term: 20
• First term: 12
• Common difference ( d ): 20 - 12 = 8

This shared increment ( d = 8 ) confirms that we are dealing with a consistent arithmetic pattern, which will be the backbone for determining any other term in the sequence.

The concept of the 'Nth term' allows you to find any term in an arithmetic sequence without listing all terms. The rule to find the nth term ( a_n ) of an arithmetic sequence is to start with the first term and continue by adding the common difference ( d ) multiplied by one less than the term position ( n - 1 ). Using the sequence from the problem, and knowing the first term is 12 and the common difference is 8, the nth term rule is:

• Start from the first term: 12
• Add ( n-1 ) times the common difference: 8

The formula becomes: 12 + (n-1)8.

Mathematical formulae for sequences take abstract concepts and make them applicable. In the context of arithmetic sequences, the formula used to find the nth term is crucial.The rule or formula derived for an arithmetic sequence can be expressed as:\[a_n = a + (n-1)d\]Where:

• a is the first term
• d is the common difference

This formula helps to easily calculate any term's exact value without needing to list all intermediate terms manually. For example, to find the 20th term ( a_{20}) in our sequence, substitute 12 for a, 8 for d, and 20 for n:\[a_{20} = 12 + (20-1)8 = 164\]Thus, the 20th term of the sequence is calculated to be 164. This shows the powerful simplicity of the mathematical formula in unraveling terms within sequences.