By applying this calculator for Arithmetic & Geometric Sequences, the n-th term and the sum of the first n terms in a sequence can be accurately obtained.

**How to use this calculator:**

- Use the dropdown menu to choose the sequence you require
- Insert the n-th term value of the sequence (first or any other)
- Insert common difference / common ratio value
- Indicate how many terms required
- Use the "Calculate" button to produce the results.

The terms consist of an ordered group of numbers or events that, being presented in a definite order, produce a sequence. To this end, an Arithmetic and Geometric approach are integral to such a calculation, being two sure methods of producing pattern-following sequences and demonstrating how patterns come to work. The differences between the two sequence types depend on whether they are arithmetic or geometric in nature. Arithmetic sequences consist of consecutive terms with a constant difference, whereas geometric sequences consist of consecutive terms in a constant ratio.

## Arithmetic Sequence

Known as either an arithmetic sequence or arithmetic progression, the defining factor is dependent on the ability to produce the next term by adding or subtracting the same value.

The common difference, which is the difference between each term – or **d**
– can be attained by taking any two pairs of bordering terms and subtracting
them.

An arithmetic sequence follows the system of: a, a + d, a + 2d, a + 3d, ... , a + nd

In this sequence: **a** is the first term and **d** the common difference.

In contrast, an arithmetic sequence with a common difference of −3 is: 2,
−1, −4, −7, 10. Choosing any two pairs of numbers from the
sequence and subtracting them gives **d**. Providing the numbers border each
other, this sum works regardless of which pair
of numbers are chosen.

−1−(2) = −3

−4−(−1) = −3

−7−(−4) = −3

Etc.

### Arithmetic Sequence Formulas

1. Terms
Formula: a_{n} = a_{1} + (n - 1)d

2. Sum
Formula: S_{n} = n(a_{1} + a_{n}) / 2

*Where:*

**a _{n}** is the n-th term of the sequence,

**a _{1}** is the first term of the sequence,

**n** is the number of terms,

**d** is the common difference,

**S _{n}** is the sum of the first n terms of the sequence.

## Geometric Sequence

Known as either as geometric sequence or geometric progression, multiplying or
dividing on each occasion to obtain a successive term produces a number sequence.
Dividing any bordering pair of terms then allows for obtaining the difference
between them, which is the common ratio – or **r**.

a, ar, ar^{2}, ar^{3}, ... , ar^{n} is the resulting form for
the geometric sequence.

In this sequence: **a** is the first term and **r** the common ratio.

In contrast, a geometric sequence of 3, 6, 12, 24, 48... produces the common ratio of 2. Taking a pair of numbers from the sequence and dividing them produces the common ratio, providing the numbers chosen border each other.

6 / 3 = 2

12 / 6 = 2

24 / 12 = 2

Etc.

### Geometric Sequence Formulas

1. Terms
Formula: a_{n} = a_{1}(r^{n-1})

2. Sum
Formula: S_{n} = a_{1}(1 - r^{n}) / (1 - r)

*Where:*

**a _{n}** is the n-th term of the sequence,

**a _{1}** is the first term of the sequence,

**n** is the number of terms,

**r** is the common ratio,

**S _{n}** is the sum of the first n terms of the sequence.

By applying this calculator for Arithmetic & Geometric Sequences, the n-th term and the sum of the first n terms in a sequence can be accurately obtained.

**How to use this calculator:**

- Use the dropdown menu to choose the sequence you require
- Insert the n-th term value of the sequence (first or any other)
- Insert common difference / common ratio value
- Indicate how many terms required
- Use the "Calculate" button to produce the results.

The terms consist of an ordered group of numbers or events that, being presented in a definite order, produce a sequence. To this end, an Arithmetic and Geometric approach are integral to such a calculation, being two sure methods of producing pattern-following sequences and demonstrating how patterns come to work. The differences between the two sequence types depend on whether they are arithmetic or geometric in nature. Arithmetic sequences consist of consecutive terms with a constant difference, whereas geometric sequences consist of consecutive terms in a constant ratio.

## Arithmetic Sequence

Known as either an arithmetic sequence or arithmetic progression, the defining factor is dependent on the ability to produce the next term by adding or subtracting the same value.

The common difference, which is the difference between each term – or **d**
– can be attained by taking any two pairs of bordering terms and subtracting
them.

An arithmetic sequence follows the system of: a, a + d, a + 2d, a + 3d, ... , a + nd

In this sequence: **a** is the first term and **d** the common difference.

In contrast, an arithmetic sequence with a common difference of −3 is: 2,
−1, −4, −7, 10. Choosing any two pairs of numbers from the
sequence and subtracting them gives **d**. Providing the numbers border each
other, this sum works regardless of which pair
of numbers are chosen.

−1−(2) = −3

−4−(−1) = −3

−7−(−4) = −3

Etc.

### Arithmetic Sequence Formulas

1. Terms
Formula: a_{n} = a_{1} + (n - 1)d

2. Sum
Formula: S_{n} = n(a_{1} + a_{n}) / 2

*Where:*

**a _{n}** is the n-th term of the sequence,

**a _{1}** is the first term of the sequence,

**n** is the number of terms,

**d** is the common difference,

**S _{n}** is the sum of the first n terms of the sequence.

## Geometric Sequence

Known as either as geometric sequence or geometric progression, multiplying or
dividing on each occasion to obtain a successive term produces a number sequence.
Dividing any bordering pair of terms then allows for obtaining the difference
between them, which is the common ratio – or **r**.

a, ar, ar^{2}, ar^{3}, ... , ar^{n} is the resulting form for
the geometric sequence.

In this sequence: **a** is the first term and **r** the common ratio.

In contrast, a geometric sequence of 3, 6, 12, 24, 48... produces the common ratio of 2. Taking a pair of numbers from the sequence and dividing them produces the common ratio, providing the numbers chosen border each other.

6 / 3 = 2

12 / 6 = 2

24 / 12 = 2

Etc.

### Geometric Sequence Formulas

1. Terms
Formula: a_{n} = a_{1}(r^{n-1})

2. Sum
Formula: S_{n} = a_{1}(1 - r^{n}) / (1 - r)

*Where:*

**a _{n}** is the n-th term of the sequence,

**a _{1}** is the first term of the sequence,

**n** is the number of terms,

**r** is the common ratio,

**S _{n}** is the sum of the first n terms of the sequence.