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A sequence is a set of terms, in a definite order, where the terms are obtained by some rule. A finite sequence ends after a certain number of terms. An infinite sequence is one that continues indefinitely. For example: 1, 3, 5, 7, … (This is a sequence of odd numbers) 1st term = 2nd term = 3rd term = 2x3–1 =1 =3 =5 nth term = 2xn–1 = 2n - 1 . . . 2x1–1 2x2–1 . . . +2 +2 NOTATION 1st term = 2nd term = 3rd term = . . . u1 u2 u3 . . . nth term = un OR 1st term = 2nd term = 3rd term = . . . u0 u1 u2 . . . nth term = un-1 FINDING THE FORMULA FOR THE TERMS OF A SEQUENCE A recurrence relation defines the first term(s) in the sequence and the relation between successive terms. For example: 5, 8, 11, 14, … u1 u2 u3 =5 = u1 +3 = u2 +3 . . . un+1 = un +3 =8 = 11 = 3n + 2 What to look for when looking for the rule defining a sequence Constant difference: coefficient of n is the difference 2nd level difference: compare with square numbers (n 2 = 1, 4, 9, 16, …) 3rd level difference: compare with cube numbers (n3 = 1, 8, 27, 64, …) None of these helpful: look for powers of numbers n-1 (2 = 1, 2, 4, 8, …) k k Signs alternate: use (-1) and (-1) -1 when k is odd +1 when k is even EXAMPLE: Find the next three terms in the sequence 5, 8, 11, 14, … EXAMPLE: The nth term of a sequence is given by x n = a) 1 __ 2 n Find the first four terms of the sequence. 1 ____ 1024 b) Which term in the sequence is ? c) Express the sequence as a recurrence relation. EXAMPLE: Find the nth term of the sequence +1, -4, +9, -16, +25, … EXAMPLE: A sequence is defined by a recurrence relation of the form: M n + 1 = aM + b. Given that M1 = 10, M2 = 20, M 3 = 24, find the value of a and the value of b and hence find M4 . This powerpoint was kindly donated to www.worldofteaching.com http://www.worldofteaching.com is home to over a thousand powerpoints submitted by teachers. This is a completely free site and requires no registration. Please visit and I hope it will help in your teaching.