# Arithmetic Sequence

Arithmetic Sequence is a type of sequence out of all sequences where each term of the sequence is related to the previous term of the sequence by a linear relation. A sequence is a collection of objects where all the terms follow an order or pattern by which the whole sequence can be identified. In the case of an Arithmetic Sequence, each term can be found by adding a constant to the preceding term of the Arithmetic Sequence, this constant sets the Arithmetic Sequence apart from the other sequences. In this article, we will explore the concept of Arithmetic Sequence and various different formulas associated with it. We will also learn about the various properties of Arithmetic Sequences.

## What is Arithmetic Sequence?

In the arithmetic sequence, the absolute difference between one term and the next term is constant. In other words, a sequence a_{1}, a_{2}, . . ., a_{n }is called an arithmetic sequence or arithmetic progression if a_{n+1 }– a_{n }= d where d is constant and it is the common difference of the sequence.

Let’s make a general example of an Arithmetic Progression as follows:

a, a+d, a+2d, a+3d, . . .Where

ais the first term of Arithmetic Sequencedis a common difference between any two consecutive terms of Arithmetic Sequence

### Examples of Arithmetic Sequence

Here are some examples of arithmetic sequences,

Example 1: Sequence of even number having difference 4 i.e., 2, 6, 10, 14, . . . ,Here in the above example, the first term of the sequence is a

_{1}=2 and the common difference is 4 = 6 -2.

Example 2: -3, 0, 3, 6, 9, 12, . . .In this sequence, the common difference is 3. Each term is obtained by adding 3 to the preceding term.

Example 3: 100, 90, 80, 70, 60, …In this sequence, the common difference is -10. Each term is obtained by subtracting 10 from the preceding term.

## Formula for Arithmetic Sequence

As we already discussed that the arithmetic sequence is a series of numbers where each number is calculated by adding a constant in the previous term. There are various formulas that are used in the Arithmetic sequence that are,

- N
^{th}Term of Arithmetic Sequence - Recursive Formula for Arithmetic Sequence
- Sum of terms in Arithmetic Sequence

### N^{th} Term of Arithmetic Sequence

Let’s consider an example of Arithmetic Sequence 6, 16, 26, 36, 46, 56, 66, . . ..

6 | 6+0.10=6 |

6+10 | 6+1.10=16 |

6+10+10 | 6+2.10=20 |

6+10+10+10 | 6+3.10=36 |

6+10+10+10+10 | 6+4.10=46 |

6+10+10+10+10+10 | 6+5.10=56 |

6+10+10+10+10+10+10 | 6+6.10=66 |

and so on. . . |

As we can see each term of this example can be represented in a similar form. Thus, the n^{th} term can be found easily by adding one less than n multiple of 10 to the first term of the sequence i.e., 6.

Thus, the n^{th} term of the given example can be generalized as **6 + (n-1)Ă—d**.

In general, this is the standard explicit formula of an arithmetic sequence whose first term is, A, end, and the common difference is D is given as follows:

a_{n}= a + (n-1)Ă—d

**Read more on ****How to Find the Nth term of Arithmetic Sequence?**

### Recursive Formula for Arithmetic Sequence

The n^{th} term of an Arithmetic Sequence can be defined recursively as the next term can always be obtained by adding a common difference to the preceding term, the following derivation can be used to illustrate the same thing.

As we know, n

^{th}term of the Arithmetic Sequence is given by

a_{n}= a + (n-1)Ă—dthus, (n-1)

^{th}term can be given by

a_{n-1}= a + (n-1-1)Ă—d

a_{n-1 }= a + (n-2)Ă—dThus,

a_{n}= a + (n-1)Ă—d

â‡’ a_{n }= a+(n-1-1+1)Ă—d

â‡’ a_{n }= a + (n-2+1)Ă—d

â‡’ a_{n }= a + (n-2)Ă—d + d

â‡’ a_{n}= a_{n-1}+ d

**Sum of terms in Arithmetic Sequence**

Let’s sequence is given as a, a+d, a+2d, a+3d, ….. a+(n-1)d.

S_{n }= (n/2)(a + l)

where,

ais the first termlis the last term of the series andnis the number of terms in the series

Replacing the last term l by thenwe get^{th}term = a + (n â€“ 1)d,

S_{n }= (n/2)(a + a + (n â€“ 1)d)

OR

S_{n }= (n/2)(2a + (n â€“ 1) x d)

## Arithmetic Series

The sum of terms of an Arithmetic Sequence is called Arithmetic Series. We use the sum of n terms formula of the arithmetic sequence to find the sum of the arithmetic series.

Suppose the first term of the arithmetic series is **a** and the common difference is **d** then the sum of the **n** term of this arithmetic series is given using the formula,

S_{n}= n/2 [2a + (n-1)d]

If the common difference of the arithmetic series is not given but the nth term of the series is given (say l). Then its sum is calculated as,

S_{n}= n/2 [a + l]

We can understand this with the help of the example,

**Example: A tree fruits five apples in the first year and at each successive year it has 2 more apples than the last year find the total apple the tree bears at the end of six years.**

**Solution:**

Apple bear by tree in first year (a) = 5

Yearly Increase in apple bear by the tree (d) = 2

Time Period = 6 years

Total apple at the end of five years in the tree.

S

_{n}= n/2 [2a + (n-1)d]â‡’ S

_{n}= 6/2(2(5) + (6 – 1)(2))â‡’ S

_{n }= 6/2 (10 + 10)â‡’ S

_{n }= 3(20)â‡’ S

_{n }= 60Thus, the apple in the tree at the end of six-year is 60 apples

**Properties of Arithmetic Sequence**

There are some properties of Arithmetic Sequence, some of which are as follows:

- If a constant is added or subtracted to each term of an Arithmetic Sequence then the resulting sequence is also an Arithmetic Sequence.
- If each term of an Arithmetic Sequence is multiplied or divided (not by 0) by a constant number. Then the resulting sequence is also an Arithmetic Sequence.
- For any three consecutive terms of an Arithmetic Sequence sum of the first and last term is always twice the middle term.
- We can observe a symmetry about the mean in the arithmetic sequence.
- An arithmetic sequence can be extended to infinity by adding a common difference to the last term.

## Difference Between Arithmetic Sequence and Geometric Sequence

The key differences between Arithmetic Sequence and Geometric Sequence are as follows:

Property | Arithmetic Sequence | Geometric Sequence |
---|---|---|

Definition | A sequence in which each term is found by adding a fixed number to the previous term. | A sequence in which each term is found by multiplying the previous term by a fixed number. |

n^{th} Term | a_{n} = a_{1} + (n-1)d | a_{n} = a_{1}r^{n-1} |

Relationship between terms | The difference between any two consecutive terms is constant. | The ratio of any two consecutive terms is constant. |

Common Difference/Ratio | d = a_{n} – a_{n-1} | r = a_{n}/a_{n-1} |

Example | 2, 5, 8, 11, 14, . . . | 3, 6, 12, 24, 48, . . . |

Sum of n Terms | S_{n} = (n/2)(a_{1} + a_{n}) | S and S |

Uses | Used to model linear relationships or changes that occur at a constant rate. | Used to model exponential growth or decay. |

**Learn more about, ****Difference between an Arithmetic Sequence and a Geometric Sequence**

**Read More,**

## Sample Problems on Arithmetic Sequence

**Problem 1: Write the first three terms in each of the following sequences defined by **

**A**_{n}=5n+2(n-1)**A**_{n}=2n +4(n-2)

**Solution :**

For A_{n}=5n+2(n-1)Put n=1, we get

a

_{1}=5.1 + 2(1-1) = 5+ 0 =5Put n=2, we get

a

_{2}=5.2+2(2-1) =10+2 =12Put n=3, we get

a

_{3}=5.3 + 2(3-1) =15 + 4 =19So first three terms are 5,12, 19.

For A_{n}=2n +4(n-2)Put n=1, we get

a

_{1}=2.1+4(1-2) =2-4 = -2Put n=2, we get

a

_{2}= 2.2+4(2-2) =4+ 0 =4Put n=3,we get

a

_{3}= 2.3 + 4(3-2) =6+4 =10So the first three terms are -2, 4, 10.

**Problem 2: Find the 20 ^{th }Term of the given expression A_{n}=(n-1)(2-n)(3+n).**

**Solution:**

For A

_{n}=(n-1)(2-n)(3+n)Put n=20 in given expression,

a

_{20 }=(20-1)(2-20)(20+3)â‡’ a

_{20 }= 19Ă—(-18)Ă—(23)â‡’ a

_{20 }= -7886.

**Problem 3: Find the sum of all natural numbers lying between 100 and 1000 (inclusive of both 100 and 1000) which are multiples of 5.**

**Solution:**

Solve: first term to be 100 and last terms is 1000 and common difference is 5.

So our formula is S

_{n}=(n/2)[2a+(n-1)Ă—d] .using a

_{n}= a_{1}+ (n-1)dâ‡’ 1000 = 100 + (n – 1)5

â‡’ 900 = (n – 1)5

â‡’ 180 = n – 1

â‡’ n = 181

Thus, there are 181 such number. Now for sum of all the 181 terms of sequence can be calculated as follows:

S

_{181 }= (181/2)[2Â·100 +(181-1)Ă—5].â‡’ S

_{181 }= (181/2)[200+180Ă—5]â‡’ S

_{181 }= (181/2)Ă—1100â‡’ S

_{181 }= 181Ă—550 = 99,550

## FAQs on Arithmetic Sequence

### Q1: Define Arithmetic Sequence.

**Answer:**

Arithmetic Sequence is defined as the sequence where each term of the sequence can be calculated by adding a constant in the preceding term of the same sequence.

### Q2: What is the Common Difference of Arithmetic Sequence?

**Answer:**

The difference between two consecutive terms of an Arithmetic Sequence is called the common difference of an Arithmetic Sequence.

### Q3: What is the n^{th} term of an Arithmetic Sequence?

**Answer:**

The nth term of an arithmetic sequence is given by the following formula:

a_{n}= a+(n-1)Ă—d

where,

ais the first term of the sequence,dis the common difference of the sequence,nis the number of terms in the sequence,

### Q4: How to find the Sum of the First n terms of an Arithmetic Sequence?

**Answer:**

We can use the following formulas to find the sum of n terms of an arithmetic sequence.

S_{n }= (n/2)(a + l)

S_{n }= (n/2)(2a + (n â€“ 1) x d)

where,

ais the first term of the sequence,dis the common difference of the sequence,lis the last term of the sequence, andnis the number of terms in the sequence,

### Q5: What is the Difference between an Arithmetic Sequence and a Geometric Sequence?

**Answer:**

In Arithmetic Sequence, the difference between any two consecutive terms is constant. Whereas in a geometric sequence, the ratio of any two consecutive terms is constant.

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