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# Arithmetic Progression or Arithmetic Sequence

- March 17, 2018
- Posted by: allexamshelps
- Category: Mathematics

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__Understanding Arithmetic Progression or Arithmetic Sequence with clarity__:

__Understanding Arithmetic Progression or Arithmetic Sequence with clarity__:

#### In Mathematics, a sequence is a list of ordered numbers.

#### This means each number in this list has an order.

#### Ex: 1, 2, 3…..N.

#### 1 is the first number; 2 is the second number; 3 is the third number.

#### In other words, a listing of events in a chronological order makes a sequence.

**Arithmetic Sequence or Arithmetic Progression:**

#### A sequence becomes an arithmetic progression or arithmetic sequence when it follows a certain pattern. This means we can find the next number/term of that sequence by adding a fixed integer. The first term is always given in normal cases.

#### (Numbers are called terms in an A.P)

#### { a }_{ 1 } { a }_{ 2 } { a }_{ 3 } { a }_{ 4 } { a }_{ 5 }

#### 1 2 3 4 5...

#### **+1 +1 +1 +1 (+1 is a Common difference (c.d))**

#### Each time we added a fixed number, +1, to obtain the next term in the above sequence.

#### { a }_{ 1 } { a }_{ 2 } { a }_{ 3 } { a }_{ 4 } { a }_{ 5 }

#### 1000 800 600 400 200

#### ** **** -200 -200 -200 -200 (-200 is a Common difference (c.d))**

#### Each time we added a fixed number, -200, to obtain the next term in the above sequence.

#### { a }_{ 1 } { a }_{ 2 } { a }_{ 3 } { a }_{ 4 } { a }_{ 5 }

#### 100 100 100 100 100

#### ** (+) 0 (+) 0 (+) 0 (+) 0 ( zero is a Common difference (c.d))**

#### Each time we added a fixed number, 0, to obtain the next term in the above sequence.

**Two things should become clear from the examples above that:**

**1). the first term is always given (in normal cases).**

**2). the common difference is also given. If it is not given, it can be found using the following formula.**

#### c.d = { a }_{ 2 } - { a }_{ 1 }; c.d = { a }_{ 3 } - { a }_{ 2 }

#### This means we subtract the preceding (which comes before) term from the succeeding (next) term to obtain the common difference.

#### (Note: when the common difference is not given directly, then at least 3 terms of the A. P should be given. Otherwise, it can cast doubts whether it is an A.P or not. See the third example above, a zero is being added to obtain the next term.

#### But if we are given a sequence of the following form:

#### { a }_{ 1 } { a }_{ 2 }

#### 100, 100…

#### Then just observing two terms, you will conclude that it is an A.P where each next term will be obtained by adding a zero, but actually; it is not an A.P. Because the next terms after the second term are:

#### 100, 100, 150, 200…

### The general form of an A.P is written in the following way:

##### a_{ 1 },\quad { a }_{ 1 }+c.d,\quad a_{ 1 }+2c.d,\quad a_{ 1 }+3c.d,\quad { a }_{ 1 }+4c.d…

#### This means; if { a }_{ 1 } is the first term, then adding the common difference (c.d) to the first term gives the second term.

#### For the third term (3^{rd} place value)

#### The first term plus (n-1)c.d **where n is the place value which is 3 here.**

#### This implies to find the 4^{th} term; the common difference is multiplied by 3, (n-1)=(4-1)=3 where n=4, then it is added to the first term.

### Finding the { n }^{ th } term of an A.P:

#### As already demonstrated, that to find the fourth term of an A.P, we use (n-1) multiplied by the common difference, and then this product is added to the first term to find the { n }^{ th } term.

**This is what makes the formula of Arithmetic Progression: so, instead of memorizing the formula, understand this logic you will never forget it.**

#### For example, if we want to find the 6^{th} term of the A.P whose three terms are given

#### { a }_{ 1 } { a }_{ 2 } { a }_{ 3 }

#### 1000 800 600

#### Then firstly find the common difference:

#### c.d = { a }_{ 2 } - { a }_{ 1 }; c.d = { a }_{ 3 } - { a }_{ 2 } = -200.

#### That is a common difference is -200.

#### Then

#### Apply the formula:

**a_{ n }=a_{ 1 }+(n-1)c.d **

**Where n=6, c.d= -200 if n=6 then you will be multiplying the common difference by 5.**

** a_6=1000+(6-1)-200**

** a_6=1000+5 (-200)**

** a_{ 6 }=1000+-1000**

**a_{ 6 }=0**

#### Hence the sixth term is zero.

#### Similarly to find the number of terms of an A.P, we write the last term **(this happens only in the case of a finite A.P)** of that A.P on the left-hand side of the above formula:

#### 5, 8, 11…32

#### a_{ L }=a_{ 1 }+(n-1)c.d

#### Where L=the last term, and **n is unknown**

**32=5+(n-1)3**

**32=5+3n-3**

**32=2+3n**

**32-2=3n**

**\frac { 30 }{ 2 } =n**

#### 10=n

#### Finite A.P:

#### If an A.P has the last term, then it is a finite, otherwise infinite.

#### Finite A.P example

#### 5, 8, 11…32

#### Where 32 is the last term denoted by a small L (l).

#### Infinite A.P

#### 1, 2, 3…N this is countable but infinite.

**Hence it is proved that in order to find out whether a sequence is an Arithmetic sequence or not we have to have two pieces of information:**

** 1) The first term, and**

** 2) The common difference.**

**And we have learned how to make the formula of Arithmetic progression used to find:**

**1). The nth term of an A.P, and**

**2) The number of terms of an A.P ( in the case of a finite A.P)**

#### (Please leave your comments, to encourage us, to let us know how good/bad we are doing it (Please support your comments with reasons)).

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