In this article, we will discuss about the free sequence alignment. Sequence alignment is a method of arranging the sequences. It has two types of sequences.

1. Arithmetic sequence and

2. Geometric sequence.

Arithmetic sequence means that, the sequence of numbers such that the difference between two consecutive members of the sequence is a constant. Geometric sequence means that, the sequence of numbers such that the ratio between two consecutive members of the sequence is a constant. The free sequence alignment formulas and example problems are given below.

Sequences formulas are given below.

**Formula for arithmetic sequence:**

n^{th} term of the sequence : a_{n} = a_{1} + (n - 1)d

Series of the sequence: s_{n} = `(n(a_1 + a_n))/2 `

**Formula for geometric sequence:**

n^{th} term of the sequence: a_{n} = a_{1} * r^{n-1}

Series of the sequence: s_{n} = `(a_1(1-r^n))/(1 - r)`

**Example problem 1:**

Find the 10^{th} term of the given series 11, 31, 51, 71,.....

**Solution:**

First term of the series, a_{1} = 11

Difference of two consecutive terms, d = 31 - 11 = 20

n = 10

The formula to find the n^{th} term of an arithmetic series, `a_n = a_1 +
(n-1)d`

So, the 10^{th} term of the series 11, 31, 51, 71,.... = 11 + (10 - 1) 20

= 11 + 9 * 20

= 11 + 180

After simplify this, we get

= 191

So, the 10^{th} term of the sequence 11, 31, 51, 71,... is 191.

**Example problem 2:**

Find out the 6^{th} term of a geometric sequence if a_{1} = 55 and the common ratio (C.R) r =
2

**Solution:**

Use the formula `a_n = a_1 * r^(n-1)` that gives the n^{th} term to find `a_6` as follows

`a_6 = a_1 * r^(6-1)`

= 55 * (2)^{5}

= 55 * 32

After simplify this, we get

= 1760

The 6^{th} term of a geometric sequence is 1760.

The above examples are helpful to study of free sequence alignment.