Sequences and nth Term

Summary and Exam Tips for Sequences and nth Term

Sequences and nth Term is a subtopic of Algebra, which falls under the subject Mathematics in the Cambridge IGCSE curriculum. In this section, you will learn how to describe the rule for continuing a sequence and find the nth term of various sequences. You will also learn to use the nth term to find terms later in a sequence and generate sequences from patterns of shapes.

• Linear Sequences: These have a general term $U_n = an + b$. By using the difference method, you can determine the values of $a$ and $b$. For example, in the sequence 6, 10, 14, 18..., the first difference is +4, leading to $a = 4$ and the nth term $U_n = 4n - 2$.

• Quadratic Sequences: These have a general term involving $n^2$. The second difference is used to find $a$. For instance, in the sequence 2, 7, 14, 23, 34, the second difference is 2, giving $a = 1$. Solving simultaneous equations helps find $b$ and $c$, resulting in the nth term $n^2 + 2n - 1$.

• Cubic Sequences: These involve $n^3$ and use the third difference to find $a$. For the sequence 4, 16, 44, 94, 172, 284..., the nth term is $n^3 + 2n^2 - n + 2$.

• Geometric Sequences: Each term is obtained by multiplying the previous term by a constant $r$, known as the common ratio. These sequences are used to create fractals like the Sierpinski triangle and the Koch snowflake.

Exam Tips

• Understand the Difference Method: Practice using the difference method to find the nth term for linear, quadratic, and cubic sequences. This is crucial for solving sequence problems efficiently.

• Solve Simultaneous Equations: Be comfortable with solving simultaneous equations, especially for quadratic and cubic sequences, to determine the coefficients of the nth term.

• Identify Sequence Types: Quickly identify whether a sequence is linear, quadratic, cubic, or geometric. This will guide you in applying the correct method to find the nth term.

• Practice with Patterns: Engage with practice questions and past papers to familiarize yourself with different sequence patterns and their nth terms.

• Review Special Sequences: Remember special sequences like square numbers, cube numbers, triangular numbers, and Fibonacci numbers, as they often appear in exams.