In this section, we'll explore how to find the nth term of different types of sequences: linear, simple quadratic, and simple cubic. This skill is crucial for identifying patterns and predicting future values in a sequence, a fundamental aspect of algebra.

**Introduction to Sequences**

A sequence is a set of numbers following a specific pattern. Understanding this pattern allows us to predict subsequent numbers and even find the formula for the nth term, which represents any term's position in the sequence.

**Linear Sequences**

Linear sequences increase or decrease at a constant rate. The nth term of a linear sequence can be found using the formula:

$\text{nth term} = a + (n-1)d$**a**is the first term of the sequence.**d**is the common difference between the terms.**n**is the term number.

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**Example 1**:

Consider: 5, 9, 13, 17...

- d = 9 - 5 = 4
- a = 5
- nth term = 5 + (n-1)4 = 4n + 1

**Example 2:**

The sequence 3, 8, 13, 18... is linear.

a) Find the nth term

b) Calculate the 20th term in the sequence.

**Solution:**

a) d = 5, a = 3.

nth term = 5n - 2

b) 20th term = (5 x 20) - 2 = 98

**Simple Quadratic Sequences**

Quadratic sequences have a second difference that is constant and use a formula in the form of:

$\text{nth term} = an^2 + bn + c$To find the values of **a**, **b**, and **c**, we need to set up equations using the first few terms of the sequence.

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**Example 1: **

Consider: 2, 6, 12, 20, 30...

- First differences: 4, 6, 8, 10...
- Second differences: 2, 2, 2...

Hence, the nth term is of the form: $n^2 + bn + c$

**Example 2:**

Find the nth term of the sequence: 4, 9, 16, 25...

**Solution:**

- First differences: 5, 7, 9
- Second differences: 2, 2
- $a = 1$, nth term = $n^2 + bn + c$ (Further calculations would be needed to find b and c).

**Simple Cubic Sequences**

**Identifying Simple Cubic Sequences**Like quadratic sequences, cubic sequences have neither a constant first nor second difference. Instead, they will have a constant**third difference**.

**Example 1: Basic Cubic Sequence**

- Sequence: 1, 8, 27, 64, 125...
- First Differences: 7, 19, 37, 61...
- Second Differences: 12, 18, 24...
- Third Differences: 6, 6, 6... (Notice the constant third difference)

**Example 2: Real-World Application**

Imagine stacking cubes to form a larger cube.

- Number of cubes used: 1, 8, 27, 64... (Represents 1³ , 2³, 3³, 4³... cubes needed for each increasing size)
- This pattern is a cubic sequence.