2. What are patterns and sequences ?
• Patterns are repetitive sequences and can be found in nature,
shapes, events, sets of numbers and almost everywhere you care to
look. For example, seeds in a sunflower, snowflakes, geometric
designs on quilts or tiles, the number sequence 0;4;8;12;16;....
• Some types number patterns and sequences are arithmetic,
geometric, triangular and Fibonacci sequences.
3. Arithmetic patterns and Sequences
• A sequence of numbers where the difference between the
consecutive numbers is constant.
• A finite portion of an arithmetic sequence is known as an finite
arithmetic sequence.
• Example
• The behavior of the arithmetic progression depends on the
common difference.
4. Geometric Sequence
• If a sequence of values follows a pattern of multiplying a fixed
amount (not zero) times each term to arrive at the following
term, it is referred to as a geometric sequence.
• The fixed amount multiplied is called the common ratio, r,
referring to the fact that the ratio (fraction) of the second term to
the first term yields this common multiple. To find the common
ratio, divide the second term by the first term.
5. An example of geometric sequence
• Find the common ratio for the
sequence
• The common ratio, r, can be
found by dividing the second
term by the first term, which
in this problem yields -
1/2. Checking shows that
multiplying each entry by -1/2
yields the next entry.
6. Square Number Sequences
• Square numbers, better known as perfect squares, are an integer which
is the product of that integer with itself. Square numbers are never
negative.
• An example of this type of number sequence could be the following:
•
• 1, 4, 9, 16, 25, 36, 49, 64, 81, …
•
• The sequence consists of repeatedly squaring of the following numbers:
1, 2, 3, 4 etc. since the 10th number of the sequence is missing, the
answer will be 102 = 100.
7. Triangular Sequence
• A triangular number sequence is generated from a pattern of dots
forming triangles
• Eg- 1, 3, 6, 10, 15, 21, 28, 36, 45, ...
• The sequence has the number of dots forming the triangles
• Like – the first triangle has 3 dots, so the first number in the
sequence will be 3
8. Fibonacci numbers
• The Fibonacci Sequence is a special series of numbers.
• This sequence was made by Leonardo pisano bogollo, also known as
Leonardo fibonacci
• The first few numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
• In this sequence, the next number is formed by adding the two numbers
before it
• For eg. 2=1+1, 3=1+2, 21=13+8
• The next number in the list above would be 21+34=55
• A longer version of the list is
• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,
4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811,
...
