Download Exercises in Number Sequences

Exercises in Number Sequences
Arithmetic Number Sequence
An arithmetic sequence goes from one term to the next by always adding (or subtracting) the
same value known as the common difference.
Examples:
A.
2, 5, 8, 11, 14, ………… add 3 at each step so the common difference is 3
B.
7, 3, -1, -5, ………….
Subtract 4 at each step so the common difference is -4
Rule:
an = a + (n – 1)(d) where “a” is the first term, “n’ is the number of the term in the
sequence, and “d” is the common difference.
Example:
C.
Use the rule to find the 4th term in the sequence in part A.
Solution: a4 = 2 + (4 – 1)(3) = 2 + (3)(3) = 11 --- does this match the 4th term?
Problems:
1. What is the next number in the arithmetic sequence 3, 11, 19, 27, 35, _____?
2. What is the 20th term in the sequence?
Geometric Number Sequence
An geometric sequence goes from one term to the next by always multiplying (or dividing) the
same value know as the common ratio.
Examples:
D.
1, 3, 9, 27 81, …………
multiple by 3 at each step so the common ratio is 3
E.
16, 8, 4, 2, 1, 1/2, ………….
Divide by 2 at each step so the common ratio is ½
Rule:
an = arn-1 where “a” is the first term, “n’ is the number of the term in the sequence, and
“r” is the common ratio.
Example:
F.
Use the rule to find the 5th term in the sequence in part A.
Solution: a5 = 1(3)(5-1) = 1(3)4 = 81 ---- does this match the 5th term in the sequence?
Problems:
3. What is the next number in the geometric sequence 3, 6, 12, 24, 48, _____?
4. Use the rule to verify the 4th term in the sequence?
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Exercises in Number Sequences
Triangular Number Sequence
This is the Triangular Number Sequence:
1, 3, 6, 10, 15, 21, 28, ...
This sequence is generated from a pattern of dots which form a triangle.
By adding another row of dots and counting all the dots we can find the next number of the
sequence:
Rule:
xn = n(n+1)/2 where “n” is the position of the triangular number in the sequence
Example:
G.
Verify the 4th triangular number is 10.
Solution: x4 = 4(4+1)/2 =4(5)/2 = 10 ---- does this match the 4th term in the sequence?
Fibonacci Sequence
The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the two numbers before it.




The 2 is found by adding the two numbers before it (1+1)
Similarly, the 3 is found by adding the two numbers before it (1+2),
And the 5 is (2+3),
and so on!
Problem:
5. What is the next Fibonacci number in the sequence above?
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Exercises in Number Sequences
Golden Ratio
Golden Section
The golden ratio is a special number approximately equal to
1.618
The actual value is 1.61803398874989484820... (etc.) the digits keep
going, with no pattern
It appears many times in geometry, art, architecture and other
areas.
if you take any two successive (one after the other) Fibonacci
Numbers, their ratio is very close to the Golden Ratio.
In fact, the bigger the pair of Fibonacci Numbers, the closer
the approximation. Let us try a few:
A
B
2
3
3
5
5
8
8 13
...
...
144 233
233 377
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B/A
1.5
1.666666666...
1.6
1.625
...
1.618055556...
1.618025751...
Exercises in Number Sequences
Images for the golden ratio
http://www.youtube.com/watch?v=fmaVqkR0ZXg
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