Sequences

1. Overview

Sequences are ordered lists of numbers (or other elements) that follow a defined pattern or rule. In IGCSE Mathematics, you'll learn to identify these patterns, continue sequences, and express the rules as algebraic formulas. This skill is crucial not only for exam success but also for understanding more advanced mathematical concepts. You will be expected to work with linear, quadratic, cubic and geometric sequences.

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Key Definitions

• Sequence: A list of numbers following a mathematical rule.
• Term: An individual number within a sequence.
• Term-to-term rule: A rule that describes how to get from one term to the next (e.g., "add 3").
• $n$: The position of a term in the sequence (e.g., for the 1st term, $n=1$; for the 10th term, $n=10$).
• $n$th term: An algebraic expression (position-to-term rule) used to calculate the value of any term based on its position $n$.
• Common Difference ($d$): The constant value added or subtracted between terms in a linear sequence.
• Common Ratio ($r$): The constant value multiplied between terms in a geometric sequence.

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Core Content

Continuing a Sequence

To continue a sequence, identify the relationship between consecutive terms.

• Arithmetic: Adding or subtracting a fixed number.
• Geometric: Multiplying or dividing by a fixed number.
• Patterns: Sometimes sequences are shown as diagrams (e.g., matchsticks forming squares).

Linear Sequences ($n$th term)

A linear sequence changes by the same amount every time. The formula is always in the form: $dn + c$.

• $d$ is the common difference.
• $c$ is the "zero-th" term (the value of the term before the first one).

Worked example 1 — Finding the nth term of a linear sequence

Question: Find the $n$th term of the sequence 5, 8, 11, 14...

1. Find the common difference ($d$): $8 - 5 = 3$ $11 - 8 = 3$ Therefore, $d = 3$. Reason: The difference between consecutive terms is constant.
2. Multiply $n$ by $d$: $3n$ Reason: This gives the 'skeleton' of the $n$th term.
3. Find the constant ($c$): $5 - 3 = 2$ Reason: Subtract the common difference from the first term to find the 'zero-th' term.
4. Combine: $3n + 2$ Reason: Combine the 'skeleton' with the constant. The $n$th term is $3n + 2$

Verification: If $n=1$, $3(1) + 2 = 5$. If $n=4$, $3(4) + 2 = 14$. Correct.

Worked example 2 — Finding a specific term in a linear sequence

Question: The $n$th term of a sequence is given by $7n - 3$. Find the 20th term of the sequence.

1. Substitute $n = 20$ into the expression: $7(20) - 3$ Reason: The question asks for the 20th term, so $n=20$.
2. Calculate: $140 - 3 = 137$ Reason: Simplify the expression. The 20th term is 137

Simple Quadratic and Cubic Sequences

Recognize these standard sequences:

• Square numbers ($n^2$): 1, 4, 9, 16, 25...
• Cube numbers ($n^3$): 1, 8, 27, 64, 125...

If a sequence is 2, 5, 10, 17..., you should notice these are just "Square numbers + 1".

• $n$th term: $n^2 + 1$.

Worked example 3 — Finding the nth term of a simple quadratic sequence

Question: Find the $n$th term of the sequence 2, 5, 10, 17...

1. Recognize the pattern: The sequence is close to the sequence of square numbers: 1, 4, 9, 16... Reason: Identify a known sequence that is similar.
2. Determine the relationship: Each term is one more than the corresponding square number. $1 + 1 = 2$ $4 + 1 = 5$ $9 + 1 = 10$ $16 + 1 = 17$ Reason: Find the difference between the given sequence and the known sequence.
3. Write the $n$th term: $n^2 + 1$ Reason: Express the relationship algebraically. The $n$th term is $n^2 + 1$

Verification: If $n=3$, $3^2 + 1 = 9 + 1 = 10$. Correct.

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Extended Content (Extended Only)

Advanced Quadratic Sequences

For sequences where the second difference is constant, the $n$th term is $an^2 + bn + c$.

Worked example 4 — Finding the nth term of an advanced quadratic sequence

Question: Find the $n$th term of the sequence 6, 13, 22, 33...

1. Find 1st differences: $13 - 6 = 7$ $22 - 13 = 9$ $33 - 22 = 11$ Reason: Calculate the difference between consecutive terms.
2. Find 2nd difference: $9 - 7 = 2$ $11 - 9 = 2$ Reason: Calculate the difference between consecutive first differences.
3. Find $a$: $2 ÷ 2 = 1$ So, $a = 1$. (Term starts with $1n^2$). Reason: The coefficient of $n^2$ is half the second difference.
4. Subtract $an^2$ from the original sequence: Original: 6, 13, 22, 33 Subtract $n^2$ (1, 4, 9, 16): Remainder: $6-1=5, 13-4=9, 22-9=13, 33-16=17$. Reason: Isolate the linear component of the quadratic sequence.
5. Find the linear rule for the remainder: 5, 9, 13, 17... Difference is 4, so $4n$. Zero-th term is $5 - 4 = 1$. Remainder rule: $4n + 1$. Reason: Find the $n$th term of the linear sequence.
6. Final $n$th term: $n^2 + 4n + 1$ Reason: Combine the quadratic and linear components. The $n$th term is $n^2 + 4n + 1$

Geometric Sequences

These have a common ratio ($r$) instead of a difference. Formula: $u_n = ar^{n-1}$ (where $a$ is the first term).

Worked example 5 — Finding the nth term of a geometric sequence

Question: Find the $n$th term of the sequence 3, 6, 12, 24...

1. Identify first term ($a$): $a = 3$ Reason: The first term in the sequence is 3.
2. Identify ratio ($r$): $6 ÷ 3 = 2$ Reason: Divide any term by its preceding term to find the common ratio.
3. $n$th term: $3 \times 2^{n-1}$ Reason: Substitute $a$ and $r$ into the formula $u_n = ar^{n-1}$. The $n$th term is $3 \times 2^{n-1}$

Worked example 6 — Finding a specific term in a geometric sequence

Question: The $n$th term of a geometric sequence is given by $2 \times 3^{n-1}$. Find the 5th term of the sequence.

1. Substitute $n = 5$ into the expression: $2 \times 3^{5-1}$ Reason: The question asks for the 5th term, so $n=5$.
2. Calculate: $2 \times 3^4 = 2 \times 81 = 162$ Reason: Simplify the expression. The 5th term is 162

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Key Equations

• Linear $n$th term: $\bf{u_n = dn + c}$
  • $d$ = common difference; $c$ = constant.
• Quadratic $n$th term: $\bf{u_n = an^2 + bn + c}$
  • $2a$ = second difference.
• Geometric $n$th term: $\bf{u_n = ar^{n-1}}$
  • $a$ = 1st term; $r$ = common ratio.

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Common Mistakes to Avoid

• ❌ Wrong: For the sequence 10, 7, 4..., saying $d = 3$.
  • ✅ Right: The sequence is decreasing, so $d = -3$. The rule is $-3n + 13$.
• ❌ Wrong: Confusing $n$ with the term value. (e.g., "Find the 10th term" means find $u_{10}$, not solve $u_n = 10$).
  • ✅ Right: If asked for the value of the first positive term in a sequence that starts negative, calculate the value (e.g., 2), do not just say it's the "$n=5$" position.
• ❌ Wrong: Forgetting to use brackets on your calculator when $n$ is negative, especially when squaring.
  • ✅ Right: When substituting a negative value for $n$ in an expression like $n^2$, use brackets: $(-5)^2 = 25$, whereas $-5^2 = -25$.
• ❌ Wrong: Stopping after finding the common difference in a linear sequence question.
  • ✅ Right: Always find the complete $n$th term expression (including the constant $c$) and then VERIFY your expression by substituting $n=1$ and $n=2$ to check that you get the first two terms of the sequence.

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Exam Tips

• Command Words: "Write down the next two terms" requires no working. "Find an expression for the $n$th term" requires full algebraic working to ensure method marks.
• Verification: Always test your $n$th term formula with $n=2$ or $n=3$ to see if it matches the sequence. If it doesn't, you have a sign error.
• Calculator Tip: Use the TABLE mode on your scientific calculator. Enter your $n$th term formula as $f(x)$ to quickly generate a list of terms to check against the question.
• Mark Loss: Students often lose marks for not showing the "differences" rows. Always write out the 1st and 2nd differences clearly when finding quadratic sequences.
• Pattern Questions: If the question involves shapes, count the items and write them as a list of numbers first. This converts a geometry problem into a simpler number sequence problem.