Page 25 - Ganit Kaushal

11. What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you
get? Can you explain it with a picture?

12. Look at the given matchstick pattern of equilateral triangles. The triangles are not separated and any
two neighboring triangles have a common matchstick.

(i) (ii) (iii)

Observe the pattern and answer the following questions:
(a) How many matchsticks are needed in the 7th figure (term) of this sequence?
(b) How many equilateral triangles will be formed with the help of 45 matchsticks?

(c) Give the formula that represents the n-th general term of this pattern/sequence.
13. Look at the given matchstick pattern of the letter ‘A’. The As are not separated and any two neighbouring
As have 2 common matchsticks.

(i) (ii) (iii)

Observe the pattern and answer the following questions:
(a) How many matchsticks are needed in the 9th figure (term) of this sequence?
(b) How many As will be formed with the help of 62 matchsticks?

(c) Give the formula that represents the nth general term of this pattern/sequence.

1.7 PATTERNS IN SHAPES (SHAPE SEQUENCES)

Shape patterns, also known as shape sequences, are among the other significant and fundamental patterns that
exist in mathematics.
Now, we demonstrate a few key shape sequences that are studied in mathematics.

1. Regular Polygons: The following figures show the shape sequence of a few regular polygons:

(Regular polygons
sequence)
Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon
2. Complete Graphs: A complete graph, denoted as K , is a type of graph where every pair of distinct
n
vertices is connected by a unique edge (line segment). In other words, in a complete graph with n vertices,
each vertex is directly connected to every other vertex by an edge.
Key characteristics of a complete graph:
• Vertices: A complete graph with n vertices is denoted as K .
n
• Edges (line segments): The total number of edges in a complete graph K is given by the
n
formula:
nn −1 )
(
E = ; where ‘n’ is the number of vertices.
2

Patterns in Mathematics 23