Sequence List of Numbers Or Terms, in Order

Warm Up Complete the pattern. 1. 15, 11, 7, 3, -1, ____, ____, ____ 2. -12, -7, -2, 3, 8, ____, ____, ____ 3. 10, 11, 13, 16, 20, 25, ____, ____, ____ 4. 2, 6, 18, 54, ____, ____, ____ 5. 96, 48, 24, 12, ____, ____, ____

Sequence – List of numbers or terms, in order. infinite – 2, 4, 6, 8, … finite – 3, 5, 7, 9 A formula that defines “nth” term is an explicit formula.

n is the position and tn is the term.

Ex. 1 If tn = 4n + 5, find the first four terms.

t1 = 4(__) + 5 =

A formula that defines a term using the previous term (tn-1) is called a recursive formula. Ex. 2 find the first four terms for

t1 = 2 tn = 2(tn-1) – 1

t1 = 2

t2 = 2(___) – 1 =

t4 = Ex. 3 Write a rule for the sequence 4, 17, 56, 173, …

Series – sum of the terms in a sequence. sequence: 2, 4, 6, 8 series: 2 + 4 + 6 + 8 =

 (sigma) – summation notation

6 Ex. 4 Find  2k ____ + ____ +____ +____ +____ +____ = ____ k1

6 Ex. 5 Find 2k 2(____ + ____ +____ +____ +____ +____) = k 1 ____

Rule: c tn  c  tn

Properties of Summation

n 4 Constant: c  n c ex. 5  ____ + ____ +____ +____ = ____ · ____ k 1 k 1 = ______n n(n 1) 7 Linear: k  ex. i  k 1 2 i1

n n(n 1)(2n 1) Quadratic: k 2  k 1 6

7 2 ex.  j  j1

5 Ex. 6 Find 4n2  2n  7  n1

5 5 5 4n2  2n  7  n1 n1 n1 (__ __ __) (__ __) 4 2  7  __  __ __

Assignment P. 696 #10-70 by 5’s Arithmetic Sequence – a sequence whose successive terms differ by the same amount (constant = d) Do #1-3 on w/s 11.2 1. 2. 3. Write an explicit formula for the nth term of each arithmetic sequence.

7. 16, 7, -2, -11, … Use formula: tn = t1 + d(n – 1)

tn = ____ + ____(n – 1)

tn = Do #8 & 9

8. d = ___ tn = 9. d = ___ tn =

Do #14 7.5, _____, _____, _____

Mean – average (middle number)

19. -12, _____, _____, _____, 16 20. 40, _____, _____,_____, _____, 100

tn = t1 + d(n – 1) tn = t1 + d(n – 1) ____ = _____ + d(___) ____ = _____ + d(___) _____ = ____d _____ = ____d _____ = d _____ = d _____, _____, ______, _____,_____, _____ th Ex. Find the 10 term of an arithmetic sequence if t3 = -5 and t6 = 16. tn = tr + d(n – r) ____ = ____ + d(___– ___ ) ____ = ____ d _____ = d

t10 = ___ + d(___– ___) or t10 = ___ + d(___– ___) t10 = ___ t10 = ___

Assignment p. 703 #9-78 by 3’s Arithmetic Series Series – Sum of the terms of a sequence (t  t ) S  n 1 n i.e. 3 + 5 + 7 + 9 + 11 =______n 2 (__ __) S  __  ___ __ 2 Do #1-3 on w/s 11.3 (on back of 11.2) 1. 2. *3.

Do #7, 13, 19 (__ __) (__ __) 7. S  __  ___ 13. S  __  ___ __ 2 __ 2

tn = t1 + d(n – 1)

t22 = _____ + ___(____)= _____

(__ __) 19. S  __  ___ __ 2

Assignment p. 710 #9-45 by 5’s Application of Arithmetic Sequences and Series see w/s #1, 4, 7, 10 Assignment : Finish w/s