# Arithmetic Sequence Calculator

Sum of first n terms
n-th term of the sequence

## Introduction of Arithmetic Sequence Calculator

A handy tool for analyzing a set of number created by adding the common difference each time, is called Arithmetic Sequence Calculator. Common difference is called constant and in formula it is denoted by d. This arithmetic sequence calculator is used to find any property of sequence.

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.

### 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.

 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.

## 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

## Formula of Arithmetic Sequence Calculator

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;

an =nth term

a1 = 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.

## 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. 1, -6, 36, -216

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

1. 2, 4, 6, 8

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

Formula for geometric sequence:

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

Here:

an= nth term

a1 =1st term

n = number of the term

r = common ratio

## 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

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

10th 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.

2nd part:

Now, find the sum of the 21st to the 50th 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