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# Illustration of Pattern

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Patterns are common in many areas of mathematics. Recurring decimals are one example. These are repeating sequences of digits which repeat infinitely. For example, 1 divided by 81 will result in the answer 0.012345679... the numbers 0-9 (except 8) will repeat forever ‚¬€ `1/81` is a recurring decimal.

Fractals are mathematical patterns that are scale invariant. This means that the shape of the pattern does not depend on how closely you look at it. Self-similarity is found in fractals.

Source: Wikipedia

Algebra numeric patterns in algebra are patterns made from numbers.

The numbers can be in a list. Any math operation in algebra (addition, subtraction, multiplication, or division) can be used to make the pattern.

Many of the patterns in algebra you see, will use addition. The same number will be added to each number in the list to make the next number in the list.

illustration of pattern:

Illustration 1:

2,4,6,8,10,12,!

Solution:

The rule here is counting by 2 using even number, so next term are 14 and 16.

Illustration 2:

1,3,5,6,9,11,13,15,!

Solution:

I think, if you guess that the rule is counting by 2 using odd numbers, you are correct.

Illustration 3:

1,1,2,3,5,8,13,21,34,55,!

Solution:

Look carefully and think!!!

If you concluded that the answer to this pattern in algebra is that you need to add number together to get the next number in line, you are correct.

1+1=2,2+1=3,3+2=5,5+3=8,!

This is a famous pattern in algebra or sequence known as the Fibonacci sequence.

Illustration 4:

A store gives a door prize to its third customer and to every 6th customer after that. Which of the following customers will get a door prize?

The 62nd

The 71st

The 79th

The 103rd

Solution:

Whenever you see a problem like this one, do not just stare at it hoping to see the pattern in algebra. Start working out the problem on paper, and after a few steps, you should be able to see the pattern in algebra. If the third customer gets a prize, as does every fifty customer who follows, the list of customer who will get prizes should be numbers.

3,8,13,18,23,28,33,!

Therefore, every number that ends in a 3 or an 8 will get a prize. Of the choices listed, only (4) ends with 3 or an 8, so it must be the answer.

Illustration 5:

Which of the following best describes the pattern in algebra of these numbers?

5/2,3,7/2,4.

Solution:

Each number is one-half more than the number before it.

The math pattern solver, from the French patron, is a type of theme of recurring events or objects, sometimes referred to as elements of a set. These elements repeat in a predictable manner. It can be a modeled in which can be used to generate things or parts of a thing. Let we discuss about the math patter solver.( Source - Wikipedia ).

The math pattern solver are deals with in following

Arithmetic Pattern

Alphabetic Pattern
Geometric Pattern

Problems in math pattern solver

Example 1 in arithmetic math pattern solver:

Determine the given sequence described by an = 4n2 + 1 and A.P.?

By using calculator we can solve the given problem:

pattern solvre calculator

When we enter the values in the pattern calculator to find the next values it finds correctly.

Now we can solve the problems in manual calculations:

Determine the given sequence described by an = 4n2 + 1 and A.P.?

Solution:

an = 2n2 + 1

a1 = 4(1)2 + 1 = 5,

a2 = 4(2)2 + 1 = 17

a3 = 4(3)2 + 1 = 37,

a4 = 4(4)2 + 1 = 65

The sequence is 5, 17, 37, 65...

Here, 17 - 5 = 12

37 - 17 = 20

65 - 37 = 28

The difference is not the same.

The given sequence is not an A.P.

Example 2 in arithmetic math pattern solver:

Find the a1 = 1, a2 = 1, an = an‚¬€1 + an‚¬€2 for n 2. Find the orders.

Solution:

a1 = 1, a2 = 1
an= an‚¬€1 + an‚¬€2 for n 2
a3= a2 + a1 = 1 + 1 = 2
a4= a3 + a2 = 2 + 1 = 3
a5= a4 + a3 = 3 + 2 = 5
Therefore the required order is: 1, 1, 2, 3, 5,...

Example 3 in arithmetic math pattern solver:

Determine the order 10, 4, ‚¬€2, ‚¬€8, ! an A.P.?
Solution:

In the given order we can determine 4 - 10 = ‚¬€2 - 4 = ‚¬€8 - (‚¬€2) = - 6
The familiar variation is ‚¬€6.

Hence the given order is an A.P.

Example 3 in arithmetic math pattern solver:

Determine 4 numbers between 3 and 38 which are in an A.P.

Solution:

Let the A.P in the form a, a + d, a + 2d,...

Here a = 3, and a + 5d = 38

€' 5d = 35, €' d = 7

The A.P. is 3, 10, 17, 24, 31, 38...

The 4 numbers between 3 and 38 are 10, 17, 24, 31

Problems in geometric progressions

Example 1 in geometric progressions math pattern solver:

The sum of the first two terms of a G.P is 2 and the sum of the first four terms is 20. Find the G.P.

Solution:

Consider the GP a, ar, ar2

Given: a + ar = 2, a(1 + r) = 2 (1)

a + ar + ar2 + ar3 = 20 €' a (1 + r) (1 + r2) = 20 (2)

By virtue of (1), (2) becomes

2(1 + r2) = 20 €' 1 + r2 = 10

€' r2 = 9 €' r = + 3

Substituting r = + 3 in (1)

If r = 3 then a = 1/2

If r = - 3 then a = ‚¬€1
1 3 9 27
The G.P is ____ _____ _____ ______ or ‚¬€1, 3, ‚¬€9, 27,...
2 2 2 2