Exploring Sequences and Progressions

A sequence is an ordered list of numbers. Each number in the list is called a term.

Examples:

Natural Numbers: 1, 2, 3, 4, 5, 6, ...
Odd Numbers: 1, 3, 5, 7, 9, 11, ...
Triangular Numbers: 1, 3, 6, 10, 15, 21, ...
Square Numbers: 1, 4, 9, 16, 25, 36, ...

In the sequence of square numbers, 1 is the first term, 4 is the second term, 25 is the fifth term and so on.

Sequences may be finite or infinite.

In the sequence of natural numbers, every number (or term) is one more than the previous number. In the sequence consisting of all the odd numbers, there is a difference of 2 between any two consecutive terms. In the sequence of triangular numbers, the difference between consecutive terms (among the first six terms) are 2, 3, 4, 5 and 6.

Arithmetic Progressions

Sequences in which the difference between consecutive terms is constant, are known as arithmetic progressions (AP). In general, an arithmetic progression (AP) can be described as

a, a + d, a + 2d, a + 3d, ..., a + (n – 1) × d

where ‘a’ is the first term and ‘d’ is the common difference.

t_n = a + (n – 1) × d is an expression for the nth term of any arithmetic progression, for some fixed values of a and d.

Sum of the First n Natural Numbers

Let S = 1 + 2 + ... + n

Then S = n + (n – 1) + ... + 1

So, 2S = n(n + 1)

$$ S = \frac{n(n + 1)}{2} $$

Geometric Progressions

A geometric progression (GP) is a list of numbers in which each term after the first term is obtained by multiplying the previous term by a fixed number. This constant factor r is called the common ratio.

The general form of a GP is

a, ar, ar², ar³, ..., ar^n–1

where ‘a’ is the first term, ‘r’ is the common ratio and tn = ar^n–1 is the n^th term.