Sequence a₁, a₂, a₃ … the sequence {a₁ + a₂ + a₃ + … + a_{n} is called a series. A series is finite or infinite according as the number of terms added is finite or infinite.

**Example:** List the first four terms of the sequence {a_{n}} = {n²}, starting with *n* = 1.

{a₁, a₂, a₃, a₄}

= {1², 2², 3², 4²}

= {1, 4, 9, 16}

**Progressions:** Sequences whose terms follow certain patterns are called progressions.

**Arithmetic Progression:** An arithmetic progression (AP) is a sequence in which terms increase or decrease regularly by the same constant. A general AP, where a is the first term of AP and d is the common difference of AP.

a, a + d, a + 2d, … a + (n – 1) d**General Term:** T_{n} = a + (n – 1) d

If the all terms of an AP are increased, decreased, multiplied and divided by the same non-zero constant, then they remain in AP.

Three consecutive numbers in AP can be taken as a – d, a, a + d.

**Example:** An arithmetic progression with initial term 4 and common difference 8, then find the 10 terms

**Solution:** T_{n} = a + (n – 1) d

a = 4, d = 8

T₁₀ = 4 + (10 – 1) 8

T₁₀ = 76

**Sum of AP:** \(Sn\,=\,\frac{n}{2}\left[ 2a+\left( n-1 \right)d \right]\,=\,\frac{n}{2}\left[ a+l \right]\)

**Arithmetic Mean:** If a, A, b are in AP, then A is called by arithmetic mean.

**Geometric Progression:** A sequence is said to be a geometric progression (GP), if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio.

a, ar, ar² ….

**Example: **1, 2, 4, … 64

This is a finite sequence with a = 1 and r = 2

The geometric progression with unlimited number of terms is called **Infinite Sequence**. It does not have a last term.1, 3, 9, 27 …

This is an infinite sequence with a = 1 and r = 3

**General Term:** T_{n} = arⁿ¯¹

**Sum of Terms:** \({{S}_{n}}\,=\,\frac{a\left( {{r}^{n}}-1 \right)}{\left( r-1 \right)}\)

**Geometric Mean:** The geometric mean (GM) of any two positive numbers a and b is given by √ab. The sequence a, G, b is GP.

G = √ab.