Types of progression (AP,GP,HP progression)

Types of progression

A progression is a sequence of numbers following a specific pattern. The most common types of progression are:

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d).

1. General form of an AP: An arithmetic sequence is written as

a, a+d, a+2d, a+3d+…

Where

• a=first term
• d=common difference
• n=number of terms
• l=last term

2. The nth term of an AP is given by

3. Sum of first n terms

The sum of the first n terms is

Or,

Example 1: Find the 10th term of the AP: 3, 7, 11, 15,…

Here,

a=3, d=7-3=4, n=10

Using:

So, the 10th term is 39.

Example 2: Find the sum of the first 20 terms of the AP: 5, 8, 11, 14,…

Here,

a=5,d=8-5=3,n=20

Using

So the sum of the first 20 terms is 670.

Geometric Progression

A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed constant called the common ratio (r).

General form of a GP

Where

• a=first term
• r=common ratio
• n=number of terms

1.nth term of a GP:

2. Sum of first n terms of a GP

3. Sum of infinite GP(|r| < 1)

If ∣r∣<1, the infinite sum converges to:

Example 1: Find the 12th term of the GP: 5, 10, 20, 40, 80,…

Here,

a=5, r=10/5=2

Using

Example 2: Find the sum of the first 12 terms of the GP: 5, 10, 20, 40, 80,…

a=5,r=10/5=2

Using

Harmonic Progression

A harmonic progression is a sequence of numbers whose reciprocals form an Arithmetic progression (AP)

General form of an HP

1. nth term of HP

2. The sum of the first $n$ terms of the harmonic progression is:

Example 1: Find the 5th term of the HP: 3, 6, 9, 12,…

Here,

a=3, d=3

First find the 5th term of the HP

A5​=a+(n−1)d=3+(5−1)×3=3+12=15

Now, the 5th term of the HP is:

​Special algebraic series

i) Sum of First n natural numbers:

ii) Sum of First n squares

iii) Sum of first n cubes