Sequences

• Tell an arithmetic sequence (constant difference) from a geometric sequence (constant ratio)
• Find the nth term of an arithmetic sequence: a_n = a_1 + (n − 1)d
• Find the nth term of a geometric sequence: a_n = a_1 × r^(n−1)
• Find the sum of the first n terms of an arithmetic sequence: n × (a_1 + a_n) / 2
• Read a recursive rule (each term defined from the one before) and generate the next few terms

A sequence is an ordered list of numbers. In an arithmetic sequence the same number d, the common difference, is added each time: 2, 5, 8, 11 has d = 3. The nth term is a_n = a_1 + (n − 1)d, so the 20th term of that sequence is 2 + 19(3) = 59. The (n − 1) matters: to reach the 20th term you take only 19 steps. In a geometric sequence each term is multiplied by the same number r, the common ratio: 3, 6, 12, 24 has r = 2. The nth term is a_n = a_1 × r^(n−1), so the 6th term is 3 × 2⁵ = 96. A ratio between 0 and 1 makes the terms shrink (80, 40, 20, 10 has r = 1/2), and a negative ratio makes the signs alternate. To decide which kind you have, subtract neighboring terms; if the differences are not constant, divide neighboring terms. The sum of the first n terms of an arithmetic sequence is n × (a_1 + a_n) / 2, the number of terms times the average of the first and last: the first 20 terms of 2, 5, 8, ... add to 20 × (2 + 59) / 2 = 610. If a question gives two non-neighboring terms of an arithmetic sequence, find d by dividing the change by the number of steps: if a_3 = 11 and a_7 = 27, then d = (27 − 11) / 4 = 4. A recursive rule defines each term from the one before, such as a_1 = 5 and a_n = 2a_(n−1) − 1, which generates 5, 9, 17, 33. Harder recursive rules change the step each time or use two earlier terms: a_1 = 5 and a_n = a_(n−1) + n² gives 5, 9, 18, 34, 59 (add 4, 9, 16, 25); and if each term is the sum of the 2 terms before it, 4, −7 continues −3, −10, −13. Write out every term in order rather than skipping ahead. A salary with the same dollar raise each year is arithmetic: $42,000 in year 1 and $46,500 in year 4 means a raise of 4,500/3 = $1,500 (3 raises, not 4), and the 4-year total is 4 × (42,000 + 46,500)/2 = $177,000. Geometric sequences are the same pattern as exponential growth and decay from the Exponents and Logarithms lesson: a fixed percent change each period is a common ratio of 1 + r or 1 − r.

• Finding the common difference (arithmetic) or common ratio (geometric) is always the first step.
• The formula uses n − 1, not n. The first term has had zero steps applied to it.
• For a short sequence, just list the terms. Writing out six terms is faster than recalling a formula you are unsure of.
• If the terms grow faster and faster, it is geometric. If they grow by the same amount each time, it is arithmetic.