Page 2 of 4. SPECIAL SEQUENCES. âAdding Previous Termsâ: Example: Fibonacci sequence......... 1 2. 2. 1. 1. 1. n n.
UNIT 4. NUMBER PATTERNS. SEQUENCES. A Sequence is a set of things (usually numbers) that are in order.
NOTATION:
x1 3 x2 5 x3 7 x4 9 . . .
THE RULE: A Sequence will have a “Rule” that gives you a way to find the value of each term. The sequence {3, 5, 7, 9 ...} starts at 3 and jumps 2 every time:
THE “nth term”: The rule for {3, 5, 7, 9...} can be written as an equation like this:
xn = 2n+1
SPECIAL SEQUENCES. “Adding Previous Terms”:
Example:
1
x1 1 x2 1 th n term x n 1 x n 2
Fibonacci sequence
1
2
3
5
1+1
1+2
2+3
3+5
8
13 8+13
21 ….. 13+21
ARITHMETIC SEQUENCES.
d = common difference
Examples:
1 +3
4
10 13 ……
7
+3
+3
25 23 21 -2
-2
-2
d = +3
common difference
+3
19 17 ……
d = -2
common difference
-2
FIND “ the nth term” USING THE FORMULA:
nth term= a1 (n 1) d
a1 first term
d= common difference
GEOMETRIC SEQUENCES.
r = ratio (common multiplier or divider) Examples:
2 x2
4
8
x2
16 32 ……
x2
:2
common multiplier
x2
64 32 16 :2
r=2
:2
8 4 ……
r=
1 2
common divider
:2
FIND “ the nth term” USING THE FORMULA:
nth term= a1 r
n 1
a1 first term
r= ratio (common multiplier or divider)
Exercise 1. Fill in the blanks the following sequences:
13
16
22
2
8
16
50
100
25
64
12.5
3.125
Exercise 2. In this sequence the rule for getting “t” is: “Multiply n by 3, then subtract 2”. Complete the table up to the 8th term.
n
1
2
3
4
5
6
7
8
t
Exercise 3. What are these sequences called?. Complete the chart below: a) 2, 4, 6, 8 …. b) 1, 3, 5, 7….. c) 1, 4, 9, 16 …. d) 1, -1, 1, -1 …. e) 5, 8, 11, 14…. f) 81, 27, 9, 3….
Exercise 4.
6
Arithmetic sequences
11
16
21
Geometric sequences
26 …..
a) Copy this sequence and continue it for three more terms. b) Write a formula for the “n th term”. c) What is the 20th term?
Neither of them
Problem 1. For each sequence of patterns: a) Predict how many lines there will be in the 4th, 5th and 6th pattern. b) Write a rule in words to calculate how many sticks are needed for each pattern. c) Write the rule in algebra (the “n th term”).
Problem 2. In an arithmetic sequence x8 = 40 and the common difference is 7. Calculate the first term of the sequence and the first ten terms’ sum.
Problem 3. In a geometric sequence the third term is 6 and the sixth one is 162. Find the “n th term” and the first twenty terms’ sum.
Problem 4. A person invests 1.000 euros in a savings account every year at a compound annual rate of 5%. What amount will he get at the end of the fifth year?.
GLOSSARY (UNIT 4) New words in English
Their meaning in Spanish
