• Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.

A list of numbers that follow a certain pattern or sequence

• Geometric Sequences

• A Geometric Sequence is made by multiplying by some value each time.

• Example:

• 2, 4, 8, 16, 32, 64, 128, 256, ...

• This sequence has a factor of 2 between each number.

• The pattern is continued by multiplying by 2 each time.

• Arithmetic Sequences

• An Arithmetic Sequence is made by adding some value each time.

• Example:

• 1, 4, 7, 10, 13, 16, 19, 22, 25, ...

• This sequence has a difference of 3 between each number.

• The pattern is continued by adding 3 to the last number each time.

Special Sequences

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, ...

This Triangular Number Sequence is generated from a pattern of dots

which form a triangle.

By adding another row of dots and counting all the dots we can find the next

number of the sequence:

Square Numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, ...

The next number is made by squaring where it is in the pattern.

The second number is 2 squared (2^2 or 2×2)

The seventh number is 7 squared (7^2 or 7×7) etc

Identifying whether the sequence is A.P, G.P, H.P

If,

a – b a A.P

b – c a

a – b a G.P

b – c b

a – b a H.P

b – c c

Arithmetic Mean A = a + b

2

G2 = AH

Geometric Mean G = √ab

A > G > H

Harmonic Mean H = 2ab

a + b

Some tips :

If first common difference is in A.P take

the General Term as ‘ax2 + bx +c’ and

determine a, b, c by solving for known

values.

If the sum of first n terms of an A.P is cn2

, then the sum of squares of these n terms is

(B) n(4n2 + 1)c2

3

(D) n(4n2 + 1)c2

6(C) n(4n2 – 1)c2

3Solution :

Sn = cn2

Tn = Sn – Sn-1

= cn2 – c(n-1)2 = c(2n – 1)

Tn2 = c2(2n – 1)2

Sn = Σ Tn

*Shortcut Method :

Put n = 1 in the

question

Above pattern is of tables at a function around which a particular number of people must sit.In order to work out how many people can sit around any arrangement of tables you must use a number of different formats.

The first of these is to draw the pattern to the sequence in the pattern you need.

Mathematical Model

#seats = slope x #of tables + y-intercept

or

Slope is the rise over the run = ---------

Y-intercept is the point where the line crosses the Y axis. =

#tables #seats

1 4

2 6

3 8

4 10

5 12

6 14

2, 4, 6, 8, 10,

____

Apply patterns

to addition…

1 + 1 = ____

5 + 3 = ____

7 + 8 = ____

• Use the shapes

for help

One heart

plus one

heart

Five smiles

plus three

smiles

Seven

stars plus

eight stars

Apply patterns to

subtraction…

3 – 2 = ____

5 – 3 = ____

12 – 9 = ____

• Use the shapes

for help Start Finish

(AGRAWAL, KUMAR, & SHAKIR) (Haggert) (Jayakumar) (www.everythingmaths.co.za) (KUMAR & SHAKIR) (learning)BibliographyAGRAWAL, A., KUMAR, A., & SHAKIR, M. B. (n.d.). MATHEMATICS.Haggert, J. (n.d.). Mathematics.Jayakumar, J. (n.d.). The Major Milestone Of Mathematics.KUMAR, A., & SHAKIR, M. B. (n.d.). Mathematics.learning, T. e. (n.d.). Introduction to pattern.www.everythingmaths.co.za. (n.d.). Everything Maths.