Ch.9 Sequences and Series Section 1 - Mathematical Patterns

Sequences Sequence: an ordered list of numbers. Each number in a sequence is a term of a sequence. We use variables with a subscript to indicate position of a number in a sequence. Ex. a 5 is the 5 th term in sequence a 1, a 2, a 3, a 4...

Cont. Subscripts usually start with 1 for the first term in the sequence. As such, a n is the n th term in a sequence. 1 st term 2 nd term 3 rd term … n-1 term nth term n+1 term... ↓ ↓ ↓ a 1 a 2 a 3 a n-1 a n a n+1

EXample Ex. 2, 4, 6, 8, 10,... The nth term is twice the value of n. So a n =2n is a general formula for the sequence. Any term in the sequence can be found. a 1 =2(1)=2, a 4 =2(4)=8, and a 10 =2(10)=20 n=1 n=2 n=3 n=4 ↓ ↓ ↓ ↓

Representing a Pattern Explicit Formula: describs the nth term of a sequence using the number n. This formula relates the terms value to the term’s placement in the sequence.

EXample 1.)Write an explicit formula for the following sequence: 1, 4, 7, 10, 13, 16, 19, 22, 25, )A sequence has an explicit formula a n =12n+3. What is term is a 12 ?

Cont. Recursive Formula: has two parts a.an initial condition: a 1 = starting number b.recursive formula (relates each term after the first term to the one before it) a 1 =b a n =a n-1 + c, where c is the pattern

EXample 133, 130, 127, 124,... a 1 =133 a n =a n-1 -3, for n>1

EXample 1.What is the recursive formula for the sequence? a.) 1, 2, 6, 24, 120, 720,... b.) 1, 5, 14, 30, 55,...