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Arithmetic Sequence Calculator
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Arithmetic Sequence Calculator is an online tool used to find nth term and sum of the sequence.

Arithmetic sequence is a sequence of numbers such that the distinction between two consecutive members of the sequence as the same common difference.

## Step by Step Calculation

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Step 1 :

The formula for the nth term an of an infinite arithmetic sequence with a common difference d and a first term a1 is given by

an = a1 + (n - 1 )d

The sum sn of the first n terms of an infinite arithmetic sequence is defined by

sn = a1 + a2 + a3 + ... + an

and is a1 is given by

sn = n (a1 + an) / 2

Where

a1 = first number of the sequence

d = common difference between the sequence

an = nth term of the sequence

n = total number of value in the sequence.

Sn = Sum of nth term of the sequence

Step 2 :

Put the values in the formulas and calculate it further.

## Example Problems

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1. ### Find the sum of all positive integers, from 27 to 4590 inclusive, which are divisible by 27.

Step 1 :

Given: Sequence of first few numbers of positive integers divisible by 27 are given by 27, 54, 81...

First term of the sequence 'a' = 27

Common difference 'd' = 27.

We need to know the rank of the term 4590.

We use the following formula for the nth term

an = a1 + (n - 1 )d

The sum sn of the first n terms of an infinite arithmetic sequence is defined by

sn = a1 + a2 + a3 + ... + an

and is a1 is given by

sn = n (a1 + an) / 2

Step 2 :

Put the values in the formula and solve it further.

an = a1 + (n - 1 )d

4590 = a1 + (n - 1 )d

Substitute a1 and d by their values

4590 = 27 + 27(n - 1)

Solve for n to obtain

n = 170

4590 is the 170th term, we can use the following formula to find sum

sn = n (a1 + an) / 2

s170 = 170 (27 + 4590) / 2 = 392445.

Answer  :

Term at position 170 = 4590

Sum of all terms till position = 392445.

2. ### Find the sum of all positive integers, from 23 to 3680 inclusive, which are divisible by 23.

Step 1 :

Given: Sequence of first few numbers of positive integers divisible by 23 are given by 23, 46, 69...

First term of the sequence 'a' = 23

Common difference 'd' = 23

We need to know the rank of the term 3680.

We use the following formula for the nth term

an = a1 + (n - 1 )d

The sum sn of the first n terms of an infinite arithmetic sequence is defined by

sn = a1 + a2 + a3 + ... + an

and is a1 is given by

sn = n (a1 + an) / 2

Step 2 :

Put the values in the formula and solve it further.

an = a1 + (n - 1 )d

3680 = a1 + (n - 1 )d

Substitute a1 and d by their values

3680 = 23 + 23(n - 1)

Solve for n to obtain

n = 160

3680 is the 160th term, we can use the following formula to find sum

sn = n (a1 + an) / 2

s160 = 160 (23 + 3680) / 2 = 296240.

Answer  :

Term at position 170 = 3680

Sum of all terms till position = 296240.

*AP and SAT are registered trademarks of the College Board.