Download 11-1 Mathematical Patterns

11-1 Mathematical Patterns
Hubarth
Algebra II
Ex. 1 Generating a Sequence
a. Start with a square with sides 1 unit long. On the right side, add on
a square of the same size. Continue adding one square at a time in
this way. Draw the first four figures of the pattern.
b. Write the number of 1-unit segments in each figure above as a sequence.
4, 7, 10, 13, . . .
c. Predict the next term of the sequence. Explain your choice.
Each term is 3 more than the preceding term.
The next term is 13 + 3, or 16.
There will be 16 segments in the next figure in the pattern.
Ex. 2 Real-World Connection
Suppose you drop a ball from a height of 100 cm. It bounces back to
80% of its previous height. How high will it go after its fifth bounce?
Original height of ball: 100 cm
After first bounce: 80% of 100 = 0.80(100) = 80
After 2nd bounce: 0.80(80) = 64
After 3rd bounce: 0.80(64) = 51.2
After 4th bounce: 0.80(51.2) = 40.96
After 5th bounce: 0.80(40.96) = 32.768
The ball will rebound about 32.8 cm after the fifth bounce.
Recursive formula defines the terms in a sequence by relating each term to the ones before it.
The pattern in example 2 was recursive because the height of the ball after each bounce was
80% of its previous height. The recursive formula that describes the ball’s height is
𝒂𝒏 = 𝟎. 𝟖𝟎𝒂𝒏−𝟏 , 𝒘𝒉𝒆𝒓𝒆 𝒂𝟏 = 𝟏𝟎𝟎
Ex. 3 Using a Recursive Formula
a. Describe the pattern that allows you to find the next term in the
sequence 2, 6, 18, 54, 162, . . . . Write a recursive formula for the
sequence.
Multiply a term by 3 to find the next term.
A recursive formula is an = an – 1 • 3, where a1 = 2.
b. Find the sixth and seventh terms in the sequence.
Since a5 = 162
a6 = 162 • 3 = 486
and a7 = 486 • 3 = 1458
c. Find the value of a10 in the sequence.
The term a10 is the tenth term.
a10 = a9 • 3
= ((a7 • 3) • 3) • 3
= (a8 • 3) • 3
= ((1458 • 3) • 3) • 3
= 39,366
Sometimes you can find the value of a term of a sequence without knowing the preceding
Term. Instead you can use the number of the term to calculate its value. A formula that
expresses the nth term in terms of n is an explicit formula.
Ex. 4 Real-World Connection
The spreadsheet shows the perimeters of regular pentagons with
sides from 1 to 4 units long. The numbers in each row form a
sequence.
A
B
C
D
1
a1
a2
a3
2
Length of Side
1
2
3
3
Perimeter
5
10
15
E
a4
4
20
a. For each sequence, find the next term (a5) and the twentieth term (a20).
In the sequence in row 2, each term is the same as its subscript.
Therefore, a5 = 5 and a20 = 20.
In the sequence in row 3, each term is 5 times its subscript. Therefore,
a5 = 5(5) = 25 and a20 = 5(20) = 100.
b. Write an explicit formula for each sequence.
The explicit formula for the sequence in row 2 is an = n. The explicit
formula for the sequence in row 3 is an = 5n.
Practice
1. Describe the patterns formed. Find the next three terms.
a. 27, 34, 41, 48, …….
b. 243, 81, 27, 9, …….
1
Add 7; 55, 62, 69
Divide by 3; 3, 1, 3
2. Describe the pattern that allows you to find the next term in the sequence 2, 4, 6, 8, 10, ….
Write a recursive formula for the sequence and find the 11th and 15th terms
𝑎𝑛 = 𝑎𝑛−1 + 2
𝑎11 = 22, 𝑎15 = 30
3. Write the first six terms in the sequence showing the areas of the squares with side lengths
of 1, 2, 3,… Then find 𝑎20 and write the explicit formula of the sequence.
1, 4, 9, 16, 25, 36
𝑎20 = 400
𝑎𝑛 = 𝑛 2