Understanding Sequences and Finding Formulas
A sequence is a set of terms in a definite order, either finite or infinite, obtained by a rule. Recurrence relations help define sequences, and finding formulas involves looking for patterns like constant difference, squared or cubed numbers comparisons, and alternations of signs.
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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 = 2 x 1 1 = 1 2nd term = 2 x 2 1 = 3 3rd term = 2 x 3 1 = 5 . . . + 2 + 2 . . . nth term = 2 x n 1 = 2n - 1
NOTATION u u u . . . 1st term = 2nd term = 3rd term = . . . 1 2 3 nth term = u n
OR u u u . . . 1st term = 2nd term = 3rd term = . . . 0 1 2 nth term = u n-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, u u u 3 = 5 = u +3 = u +3 . . . 1 = 8 = 11 1 2 2 u n+1 = u +3 n = 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= 1, 4, 9, 16, ) 3rd level difference: compare with cube numbers (n= 1, 8, 27, 64, ) None of these helpful: look for powers of numbers (2 = 1, 2, 4, 8, ) Signs alternate: use (-1) and (-1) 2 3 n - 1 k k -1 when k is odd +1 when k is even
EXAMPLE: Find the next three terms in the sequence 5, 8, 11, 14,
EXAMPLE: 1__ 2n n The nth term of a sequence is given by x = n n a) Find the first four terms of the sequence. 1 ____ b) Which term in the sequence is ? 1024 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 = aM + b. Given that M = 10, M = 20, M = 24, find the value of a and the value of b and hence find M . 4 n + 1 1 2 3
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