Sequences & Series

Pre-CalculusLesson 9.1

Infinite Sequence: A sequence without bound - -

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … ? (what’s next 2 terms)

We use the letter, a, with a subscript to represent function values of a sequence.

1, 1, 2, 3, 5, 8….

a1 = 1 a2 = 1 a3 = 2 a4 = 3 a5 = 5 a6 = 8

Finite Sequence: Find the first n terms only.

an represents the nth term of a sequence…. The entire sequence is represented by an

1. Find the first four terms of the sequence given by an = 3n – 2

2. Find the first four terms of the sequence given by: an = 3 + (-1)n

3. Find the first four terms of the sequence given by: an =

[2nd] [Stat] OPS #5[Seq]

Seq (function, variable, first term, last term)

Hit the right arrow key if all values are not visible.

SEQUENCES IN YOUR CALCULATOR

OR…. Change the mode on calculator to seq Plug in equation beside u(n) Graph Use trace to toggle through values

A recursion formula defines the nth term (an) of a sequence as a function of the previous term (an-1)

4. Find the first four terms of the sequence where a1 = 5 and an = 3an-1 + 2, where n>2.

5. Find the first four terms of the sequence where a1 = 3 and an = 2an-1 + 5, where

n>2.

Factorial NotationProducts of consecutive positive integers; the integers decrease by one…..

Example: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

6. 2 3!∙ 7. (2 3)!∙

8. List the first five terms of the sequence given by

an = Begin with n = 0

[Math] PRB #4 [!}Factorial in the calculator

Evaluate each:9. 10.

n

iia

1

4

1

3i

i

3

0 !

1

j j

Summation Notation

Compact notation for expressing the first n sums of a sequence a1 + a2 + a3 +… an =

11. 12.

7

4

5)2(k

k

5

1

3j

Expand and evaluate.

13. 14.

[2ND] [Stat] Math #5 [sum]

Then enter sequence… Sum(seq(function, variable, first term, last term))

Summations in your calculator

Express each sum using summation notation.

15. 13 + 23 + 33 + … + 73

16. 23 + 24 + … + 27

Properties of Sums

n

i

n

iii acca

1 1

n

i

n

i

n

iiiii baba

1 1 1

)(

n

i

n

i

n

iiiii baba

1 1 1

)(

Many applications involve the sum of an infinite sequence. Such a sum is called a series. To find the sum of a series, you would notice and extending pattern…17. Find the third partial sum and the sum of

n

ii

1 10

3