Number Sequences and patterns

By Swapneel, Sumesh, Rahul, Joel and Ahmed

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.

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.

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.

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.

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.

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

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, ...

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