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

The sum of the first n terms:

S_n=½n(a+ℓ)
(see this diagram)

S_n=½n(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)–a_n=d. For an arithmetic sequence:

a_n=a₁+(n-1)d=a_(n-1)+d.
💎 Geometric Sequence
Geometric sequence a, a⋅r, a⋅r², a⋅r³, …

The nth term is a_n=a⋅r^n-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⋅r^n-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=±√a⋅c

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

a_n=a₁⋅r^(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 S_n represents the nth partial sum, the sum of the first n terms of a sequence, then

S_n=∑ⁿ_k=1 a_k =a₁+a₂+a₃+⋯+a_n
💎 Arithmetic Series
Arithmetic series a+(a+d)+(a+2d)+(a+3d)+…

The sum of the first n terms is

S_n=½n(2a+(n-1)d),
where a is the first term and d is the common difference. This can also be written S_n=½n(a+ℓ), where ℓ is the nth term a_n.

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
S_n=½n(a₁+a_n) or S_n=½n(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⋅r^n-1
where a is the first term, r is the common ratio, and n is the number of terms. The nth term is a⋅r^n-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