Notes from Toppers
Arithmetic, Geometric, and Harmonic Progressions
1. Arithmetic Progressions (AP)

Definition: An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is a constant. The constant difference is called the common difference of the AP.

Explicit formula for nth term: $$T_n = a + (n  1)d$$ Where,

$$a $$is the first term

$$d$$ is the common difference, and

$$n$$ is the number of the term

Properties of AP:

The sum of two terms equidistant from the beginning and the end of an AP is always the same and equal to the sum of the first and the last terms.

The sum of n terms of an AP is given by $$S_n = \frac{n}{2}[2a + (n  1)d]$$
2. Geometric Progressions (GP)

Definition: A geometric progression (GP) is a sequence of numbers in which the ratio of any two consecutive terms is a constant. The constant ratio is called the common ratio of the GP.

Explicit formula for the nth term: $$T_n = a r^{n1}$$ Where,

$$a$$ is the first term

$$r$$ is the common ratio, and

$$n$$ is the number of the term

Properties of GP:

The product of two terms equidistant from the beginning and the end of a GP is always the same and equal to the product of the first and the last terms.

The sum of n terms of a GP is given by $$S_n = \frac{a(1  r^n)}{1  r}$$ for r ≠ 1, where r is the common ratio.

For an infinite geometric series with r < 1, the sum is given by $$S_∞ = \frac{a}{1  r}$$
3. Harmonic Progressions (HP)

Definition: A harmonic progression (HP) is a sequence of numbers in which the reciprocals of the terms form an arithmetic progression.

Explicit formula for the nth term: $$T_n = \frac{1}{a + (n  1)d}$$ Where,

$$a$$ is the first term

$$d$$ is the common difference of the arithmetic progression formed by the reciprocals of the terms, and

$$n$$ is the number of the term

Properties of HP:

The sum of two terms equidistant from the beginning and the end of an HP is always the same and equal to the sum of the first and the last terms.

The sum of n terms of an HP is given by $$S_n = \frac{n}{2}[a + \frac{1}{n} ]$$
4. Applications of Progressions:

Applications in reallife situations, such as:

Interest calculations

Population growth

Radioactive decay

Problems involving finding the sum of a finite or infinite series.

Problems involving finding the nth term or general term of a series.

Problems involving finding the sum of a series to a given accuracy.