**Arithmetic And Geometric Series Formulae**

💎 S**equences 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 *a*_{1}, *a*_{2}, *a*_{3}, … .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*+2*d*, *a*+3*d*, …

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*=½

_{n}*n*(

*a*+

*ℓ*)

(see this diagram)

*S*=½

_{n}*n*(2

*a*

_{1}+(

*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}*a*

_{1}+(

*n*-1)

*d*=

*a*

_{(n-1)}+

*d*.

💎

**Geometric Sequence**

**Geometric sequence**

*a*,

*a⋅r*,

*a⋅r*

^{2},

*a⋅r*

^{3}, …

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}*a*

_{1}⋅

*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}*a*

_{1}+

*a*

_{2}+

*a*

_{3}+⋯+

*a*

_{n}💎

**Arithmetic Series**

**Arithmetic series**

*a*+(

*a*+

*d*)+(

*a*+2

*d*)+(

*a*+3

*d*)+…

The sum of the first *n* terms is

*S*=½

_{n}*n*(2

*a*+(

*n*-1)

*d*),

where

*a*is the first term and

*d*is the common difference. This can also be written

*S*=½

_{n}*n*(

*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*+2

*d*)+(

*a*+3

*d*)+⋯+(

*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*=½

_{n}*n*(

*a*

_{1}+

*a*) or

_{n}*S*=½

_{n}*n*(2

*a*+(

*n*-1)

*d*)

💎 **Geometric Series**

**Geometric series** *a*+*a⋅r*+*a⋅r*^{2}+*a⋅r*^{3}+⋯

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⋅r*

^{2}+

*a⋅r*

^{3}+⋯+

*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