**What is Arithmetic Sequence?**

Numbers sequence, in which the diffference is always constant. It is also known as arithmetic progression. The difference is the second minus the first.

The sequence of 1, 3, 5, 7, 9, 11, ... is an arithmetic progression with common difference of 2. User must not confuse it with mean values and significant values. For learning & calculations of mean values, use Mean Calculator. For learning & calculating significant values, try Sig Fig Calculator.

**What is Arithmetic Sequence formula?**

Here we will understand the general form of an arithmetic sequence.

**First term:**

$$a_1$$

**Second term:**

$$a_2 = a_1 + d$$

**Third term:**

$$a_3 = a_1 + 2d$$

**Fourth term:**

$$a_4 = a_1 + 3d$$

**Fifth term:**

$$a_5 = a_1 + 4d$$

Arithmetic sequence formula for the nth term:

$$a_n = a_1 + (n – 1 ) d$$

Here;

a_{n} =nth term

a_{1} = 1st term

n = term number

d = the common difference

If you know any of three values, you can be able to find the fourth.

Our calculator will be helpful to find the arithmetic series by the following formula.

$$S = \frac{n}{2} * (a1 + a)$$

By putting arithmetic sequence equation for the nth term,

$$S = \frac{n}{2} * [ a1 + a1 + (n-1) d]$$

And finally it will be:

$$S = \frac{n}{2} * [ 2a1 + (n-1) d]$$

Now, this formula will provide help to find the sum of an arithmetic sequence. The distance formula has different concepts than arithmetic sequenc formula. for learning distance formula equation, use Distance Formula Calculator.

**Difference between Arithmetic Sequence and Series**

In this paragraph, we will learn about the difference between arithmetic sequence and series sequence, along with the working of sequence calculator and series calculator.

Arithmetic sequence is simply the set of objects created by adding the constant value each time while arithmetic series is the sum of n objects in sequence. So the arithmetic sequence calculator finds that specific value which will be equal to the first value plus constant. The series calculator helps to find out the sum of objects of a sequence. Look at the following numbers.

Sequences | Series |
---|---|

Set of numbers with commas | Set of numbers with plus sign |

2, 4, 6, 8, 10, 12, 14, 16, 18… | 2 + 5 + 8 + 18 + 21 + 23 + 25 … |

9, 7, 0, -3, -6, -9, - 12, - 15, -18…. | |

40, 40.1, 40.2, 40.3, 40.4, 40.5… |

Arithmetic sequence is also called arithmetic progression while arithmetic series is considered partial sum

**What is Geometric Sequence?**

Unlike arithmetic, in geometric sequence the ratio between consecutive terms remains constant while in arithmetic, consecutive terms varies.

Example:

Determine the geometric sequence, if so, identify the common ratio

- 1, -6, 36, -216

Answer: Yes, it is a geometric sequence and the common ratio is 6.

- 2, 4, 6, 8

Answer: It is not a geometric sequence and there is no common ratio.

**What is Geometric Sequence formula?**

$$a_n = a_1 * r n-1$$

Here:

a_{n}= nth term

a_{1 }=1^{st} term

n = number of the term

r = common ratio

The arithmetic equations are written on specific notations, for deep learning & understanding of scientific notation you can use Scientific Notation Calculator.

**What is Arithmetic Sequence Calculator?**

Our sum of series calculator or arithmetic series calculator is an online tool which you can find on Google. The arithmetic sequence calculator uses arithmetic sequence formula to find sequence of any property.

Actually, the term “sequence” refers to a collection of objects which get in a specific order. Objects might be numbers or letters, etc. but they come in sequence. Objects are also called terms or elements of the sequence for which arithmetic sequence formula calculator is used.

### Explanation:

To understand an arithmetic sequence, let’s look at an example. Every day a television channel announces a question for a prize of $100. If anyone does not answer correctly till 4th call but the 5th one replies correctly, the amount of prize will be increased by $100 each day.

Suppose they make a list of prize amount for a week, Monday to Saturday. As the contest starts on Monday but at the very first day no one could answer correctly till the end of the week. Sequence and combinations are not same concepts, you can learn about combination values while using our Combination Calculator.

Monday | $100 |

Tuesday | $200 |

Wednesday | $300 |

Thursday | $400 |

Friday | $500 |

Saturday | $600 |

Here prize amount is making a sequence, which is specifically be called arithmetic sequence. To find the next element, we add equal amount of first.

**How to use Arithmetic Sequence Calculator?**

All sum of a series calculator who uses arithmetic sequence formula are accurate. Below are some of the example which a sum of arithmetic sequence formula calculator uses.

## Example 1:

**Given: 39, 35, 31, 27, 23….**

$$\text{Find:}\;a_{32}$$

**Solution:**

$$a_1=39,\;d=-4\;\text{and}\;n=32$$

$$a_n = a_1 + ( n - 1 ) d$$

$$a_32 = 39 + (32 - 1) (-4)$$

$$a_32 = 85$$

## Example 2:

**Given:**

$$a_{10} = 3.25$$

$$a_{12} = 4.25$$

$$\text{Find:}\;a_1$$

**Solution:**

$$a_1=3.25$$

$$a_3=4.25$$

$$n=3$$

$$a_n=a_1+(n-1)d$$

$$4.25=3.25+(3-1)d$$

$$d=0.5$$

## Example 3:

Let us know how to determine first terms and common difference in arithmetic progression.

The third term in an arithmetic progression is **24**.

The tenth term is **3**.

Find the first term and the common difference

**Solution:**

$$\text{General formula for the nth term}$$

$$ = a_n =a_1+(n-1)d$$

**3rd term**

$$\text{equation 1:}\;24=a+2d$$

**10 ^{th} term:**

$$\text{equation 2:}\;3=a+9d-$$

$$21=-7d$$

$$\text{So, d}\;=\;\frac{21}{-7}=-3$$

**To find “a”, we will use equation 1**

$$24=a+2d$$

$$24=a+2(-3)$$

$$24=a+(-6)$$

$$\text{So,}\;a=24+6=30$$

So the first term is 30 and the common difference is -3.

**2 ^{nd} part:**

**Now, find the sum of the 21^{st} to the 50^{th} term inclusive**

**There are different ways to solve this but one way is to use the fact of a given number of terms in an arithmetic progression is $$\frac{1}{3}\;n(a+l)$$ Here, “a” is the first term and “l” is the last term which you want to find and “n” is the number of terms. In this case first term which we want to find is 21st so**

$$a_{21} = 30 + 20 (-3) = -30$$

$$a_{50} = 30 + 49 (-3) = -117$$

By putting values into the formula of arithmetic progression

$$\frac{1}{2}\;n(a + l)$$

$$\frac{1}{2}*30*(-30+(-117))$$

$$=-2205$$

So **-2205** is the sum of 21st to the 50th term inclusive.

Hope so this article will be helpful to understand the working of arithmetic calculator. We also have Remainder Calculator from which you can find the remaining values. We also have Rounding Calculator from which you can round long values easily.

Please give us the review and feedback so we could further improve.