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:

• 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:

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.