# Sequence

How to find the nth term of the given sequence by finding its pattern: 2 examples and their solutions.

## Definition

A sequence is a set of numbers in order.

a_{1} is the first term.

a_{2} is the second term.

a_{3} is the third term.

a_{4} is the fourth term.

a_{n} is the nth term.

## Example2, 3, 4, 5, 6, ..., a_{n} = ?

Find a pattern from the given terms.

Inductive Reasoning

The first term, a_{1}, is 2.

2 = 1 + 1

So a_{1} = 1 + 1.

The second term, a_{2}, is 3.

3 = 2 + 1

So a_{2} = 2 + 1.

The third term, a_{3}, is 4.

4 = 3 + 1

So a_{3} = 3 + 1.

The fourth term, a_{4}, is 5.

5 = 4 + 1

So a_{4} = 4 + 1.

The fifth term, a_{5}, is 6.

6 = 5 + 1

So a_{5} = 5 + 1.

a_{1} = 1 + 1

a_{2} = 2 + 1

a_{3} = 3 + 1

a_{4} = 4 + 1

a_{5} = 5 + 1

Then

a_{n} = n + 1.

So a_{n} = n + 1.

## Example1, 4, 9, 16, 25, ..., a_{n} = ?

Find a pattern from the given terms.

The first term, a_{1}, is 1.

1 = 1^{2}

So a_{1} = 1^{2}.

The second term, a_{2}, is 4.

4 = 2^{2}

So a_{2} = 2^{2}.

The third term, a_{3}, is 9.

9 = 3^{2}

So a_{3} = 3^{2}.

The fourth term, a_{4}, is 16.

16 = 4^{2}

So a_{4} = 4^{2}.

The fifth term, a_{5}, is 25.

25 = 5^{2}

So a_{5} = 5^{2}.

a_{1} = 1^{2}

a_{2} = 2^{2}

a_{3} = 3^{2}

a_{4} = 4^{2}

a_{5} = 5^{2}

Then

a_{n} = n^{2}.

So a_{n} = n^{2}.