Repeating Patterns

Introduction on repeating patterns:

Patterns are the part of once life. We usually follow the daily repeating patterns. Our daily routine is bounded by the repeating patterns. Waking in the morning daily at same time, Breakfast and leaving for School, office, or business the time as what we did yesterday.  Again, in our workplace repeated type of work followed by repeated activities in the evening. This is how our daily activities are all repeated patterns based.

Importance of Repeating Patterns in Mathematics and Figures:

The most common area in which the repeated patterns follows is from the topic Sequence and Series. Where, the pattern has the common difference or the common multiple, which makes the next term. But here we also look the pattern in different shapes.

In figures we can have  repeated patterns such as

An angle
A triangle
A parallelogram
A pentagon
___________ What is next
Here the pattern is of increasing the sides by 1 in next figure. Last figure is Pentagon of 5 sides So next will be figure of 6 sides and that will be Hexagon
Example:

What will the next three terms of the repeated patterns sequence?

46, 39, 32, 25, 18, _____, _____, ______?

Solution:

The two consecutive terms are having a difference of -7 {39 – 46 = -7 = 18 – 25}

Hence, following the same pattern we will have the next three terms of the repeated patterns series

Sixth term = 18 – 7 = 11

Seventh term = 11 – 7 = 5

Eighth term = 5 – 7 = - 2

Different Repeating Patterns Sequences:

Question:

Follow the repeating  pattern in the following series and answer the question?

P = 2, 5, 10, 17, 26 . . .

1.         What is the sum of the next two terms?

2.         What is the 10th term of the sequence?

Solution:

First term is 2 and the next term is 5. In these, we have to search the repeated pattern as the difference in the following term is increasing with the number of term.

We can introspect the terms, the first term can be written

2 = 1 + 1 = 12 + 1

5 = 4 + 1 = 22 + 1

10 = 9 + 1 = 32 + 1

And so on

This means the Tn = n2 + 1

1.         Sum of next two terms = T6 + T7 = (62 + 1) + (72 + 1)

= 37 + 50 = 87

2.         The 10th term = 102 + 1 = 100 + 1 = 101

I have recently faced lot of problem while learning what are corresponding angles, But thank to online resources of math which helped me to learn myself easily on net.

Questions for Practice on Repeating Patterns

1.    Fill the missing term

·         1, 2, 2, 4, ____ , 32

·         1, 5, 4, 7, 7, 9, ____, 11

·         1, 8, 27, _____, 125, 216

·         2, 6, 12, 20, _____, 42

2.    If the 3rd term of an arithmetic progression is same as the 9th term of the other progression, If the ratio of their first terms is 3 : 4 and the ratio of the common difference is 4 : 7. What is the ratio of the 5th term of both the sequences?