# Arithmetic And Geometric Series Formulae

Arithmetic And Geometric Series Formulae

💎 Sequences and Series
A sequence is a set of terms in a defined order with a rule for obtaining each term.
A series is a formed when terms of a sequence are added.

A sequence is a set of numbers written in a given order. Each term of a sequence is associated with the positive integer that specifies its position in the ordered set. A finite sequence is a function whose domain is the set of integers {1, 2, 3, … , n}. An infinite sequence is a function whose domain is the set of positive integers. The terms of a sequence are often designated as a1, a2, a3, … .The formula that allows any term of a sequence except the first to be computed from the previous term is called a recursive definition.

💎 Arithmetic Sequence
Arithmetic sequence a, a+d, a+2d, a+3d, …

The nth term is an=a+(n-1)d, where a is the first term and d is the common difference.
(a) A sequence of numbers T(1), T(2), T(3), …. T(n), is called an arithmetic sequence if:

d=T(2)-T(1)=T(3)-T(2)=…=T(n)-T(n-1)

where d is a constant known as the common difference.
(b) The first term =a=T(1)

The last term ==T(n)
(c) The general term: T(n)=a+(n-1)d

The sum of the first n terms:

Snn(a+)
(see this diagram)

Snn(2a1+(n-1)d)

(d) If a, b and c are any three consecutive terms of an arithmetic sequence, the middle term b is called the arithmetic mean of a and c, then b=½(a+c).

An arithmetic sequence is a sequence such that for all n, there is a constant d such that a(n+1)an=d. For an arithmetic sequence:

an=a1+(n-1)d=a(n-1)+d.

💎 Geometric Sequence
Geometric sequence a, a⋅r, a⋅r2, a⋅r3, …

The nth term is an=a⋅rn-1, where a is the first term and r is the common ratio.
(a) A sequence of numbers T(1), T(2), T(3), …. T(n), is called an geometric sequence if:

, where r is a constant known as the common ratio.
(b) The general term: T(n)=a⋅rn-1
The sum of the first n terms:

, where r≠1.
(c) The sum to infinity of an infinite geometric series:

, where r must be in the range: -1<r<1. (d) If a, b and c are any three consecutive terms of an geometric sequence, the middle term b is called the geometric mean of a and c, then b=±√ac

A geometric sequence is a sequence such that for all n, there is a constant r such that a(n+1)/an=r. For an arithmetic sequence:

an=a1r(n-1)=a(n-1)r

A series is the indicated sum of the terms of a sequence. The Greek letter Σ is used to indicate a sum defined for a set of consecutive integer.

If Sn represents the nth partial sum, the sum of the first n terms of a sequence, then

Sn=∑nk=1 ak =a1+a2+a3+⋯+an

💎 Arithmetic Series
Arithmetic series a+(a+d)+(a+2d)+(a+3d)+…

The sum of the first n terms is

Snn(2a+(n-1)d),

where a is the first term and d is the common difference. This can also be written Snn(a+), where is the nth term an.

In an Arithmetic Sequence, to go from one term to the next you add on the same (constant) number. This number is called the Common Difference.
This is a series of the form

a+(a+d)+(a+2d)+(a+3d)+⋯+(a+(n-1)d)

where a is the first term, d is the common difference and n is the number of terms. The nth term is a+(n-1)d.
The sum to n terms is
Snn(a1+an) or Snn(2a+(n-1)d)

💎 Geometric Series
Geometric series a+a⋅r+a⋅r2+a⋅r3+⋯

The sum of the first n terms is

where a is the first term and r is the common ratio.

In a Geometric Sequence, each term is a (constant) multiple of the previous term. This multiple is called the Common Ratio.

This is a series of the form

a+a⋅r+a⋅r2+a⋅r3+⋯+a⋅rn-1

where a is the first term, r is the common ratio, and n is the number of terms. The nth term is a⋅rn-1.
For a Geometric series, the sum to n terms is

If -1<r<1 then a Geometric Series may be summed to infinity. The sum is

For a geometric series:

For a geometric series, if |r|<1 and n approaches infinity:

As n approaches infinity:

The number e is an irrational number.
Let you read the page of Evaluating Euler’s Number and Pi π with Series