Sequences and Series

Sequence a₁, a₂, a₃ … the sequence {a₁ + a₂ + a₃ + … + an 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 {an} = {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) dArithmetic ProgressionGeneral Term: Tn = 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: Tn = 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.

\(A\,=\,\frac{\left( a\,+\,b \right)}{2}\)

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.Geometric Progression1, 3, 9, 27 …

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

General Term: Tn = 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.