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An arithmetic sequence is one where each term is obtained by adding a fixed value to the previous term. This fixed value is called the common difference, denoted by d.
u_n = u_1 + (n - 1)d, where u_1 is the first term and d is the common difference.
For example, the sequence 3, 7, 11, 15, ... has a first term u_1 = 3 and a common difference d = 4. The 10th term would be u_10 = 3 + (10 - 1)(4) = 3 + 36 = 39.
Students often write u_n = u_1 + nd instead of u_n = u_1 + (n-1)d. Remember: the “minus 1” accounts for the fact that you add d only (n-1) times to get from the 1st term to the nth term.
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