# Sequence

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

## Definition

### Definition

A sequence is a set of numbers in order.

a1 is the first term.
a2 is the second term.
a3 is the third term.
a4 is the fourth term.

an is the nth term.

## Example 1

### Solution

Find a pattern from the given terms.

Inductive Reasoning

The first term, a1, is 2.

2 = 1 + 1

So a1 = 1 + 1.

The second term, a2, is 3.

3 = 2 + 1

So a2 = 2 + 1.

The third term, a3, is 4.

4 = 3 + 1

So a3 = 3 + 1.

The fourth term, a4, is 5.

5 = 4 + 1

So a4 = 4 + 1.

The fifth term, a5, is 6.

6 = 5 + 1

So a5 = 5 + 1.

a1 = 1 + 1
a2 = 2 + 1
a3 = 3 + 1
a4 = 4 + 1
a5 = 5 + 1

Then
an = n + 1.

So an = n + 1.

## Example 2

### Solution

Find a pattern from the given terms.

The first term, a1, is 1.

1 = 12

So a1 = 12.

The second term, a2, is 4.

4 = 22

So a2 = 22.

The third term, a3, is 9.

9 = 32

So a3 = 32.

The fourth term, a4, is 16.

16 = 42

So a4 = 42.

The fifth term, a5, is 25.

25 = 52

So a5 = 52.

a1 = 12
a2 = 22
a3 = 32
a4 = 42
a5 = 52

Then
an = n2.

So an = n2.