Sequences

A sequence is an ordered list of numbers following a rule. This lesson covers term-to-term rules, the nth term of arithmetic sequences, quadratic sequences (Higher), and geometric sequences (Higher), all as required by the Edexcel GCSE (1MA1) specification.

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Key Vocabulary

Term	Meaning
Sequence	An ordered list of numbers
Term	Each number in a sequence
Term-to-term rule	A rule connecting one term to the next
Position-to-term rule (nth term)	A formula giving any term from its position number n
Arithmetic sequence	A sequence with a constant difference between consecutive terms
Common difference (d)	The amount added each time in an arithmetic sequence
Geometric sequence	A sequence with a constant ratio between consecutive terms [H]
Common ratio (r)	The number multiplied each time in a geometric sequence [H]
Quadratic sequence	A sequence whose nth term contains $n^{2} [H]$n2[H]

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Term-to-Term Rules

Worked Example 1

Write the next three terms: 3, 7, 11, 15, …

The rule is "add 4" each time. Next terms: 19, 23, 27

Worked Example 2

Write the next three terms: 2, 6, 18, 54, …

The rule is "multiply by 3" each time. Next terms: 162, 486, 1458

Worked Example 3

A sequence is defined by: $u_{1} = 5$u1​=5, $u_{n}$un​₊$_{1} = 2u_{n} - 3$1​=2un​−3. Find the first 5 terms.

$u_{1} = 5$u1​=5 $u_{2} = 2(5) - 3 = 7$u2​=2(5)−3=7 $u_{3} = 2(7) - 3 = 11$u3​=2(7)−3=11 $u_{4} = 2(11) - 3 = 19$u4​=2(11)−3=19 $u_{5} = 2(19) - 3 = 35$u5​=2(19)−3=35

First 5 terms: 5, 7, 11, 19, 35

Identifying the Type of Sequence

flowchart TD
    SEQ[Given sequence of terms] --> D1[Compute first differences]
    D1 --> Q1{First differences constant?}
    Q1 -->|Yes| ARITH["Arithmetic sequence<br/>nth term = a + n minus 1 times d"]
    Q1 -->|No| D2[Compute second differences]
    D2 --> Q2{Second differences constant?}
    Q2 -->|Yes| QUAD["Quadratic sequence H<br/>n squared coefficient = half second diff"]
    Q2 -->|No| RATIO[Compute ratio of consecutive terms]
    RATIO --> Q3{Ratio constant?}
    Q3 -->|Yes| GEOM["Geometric sequence H<br/>nth term = a times r to the n minus 1"]
    Q3 -->|No| OTHER["Special: Fibonacci, recurrence,<br/>or non-standard pattern"]

This decision tree is the standard Edexcel approach for the question "find the nth term": always test for a linear pattern first (constant first difference), then quadratic (constant second difference), then geometric (constant ratio).

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The nth Term of an Arithmetic Sequence

An arithmetic sequence has a constant common difference d.

nth term = a + (n − 1)d or equivalently nth term = dn + (a − d)

where a = first term, d = common difference.

Finding the nth Term — Step by Step

1. Find the common difference d (second term − first term).
2. The nth term starts with dn.
3. Adjust: compare dn with the actual first term to find the constant.

Worked Example 4

Find the nth term of: 5, 8, 11, 14, …

d = 3, so the nth term starts with 3n.

When n = 1: 3(1) = 3, but the first term is 5, so we need to add 2.

nth term = 3n + 2

Check: n = 4: 3(4) + 2 = 14 ✓

Worked Example 5

Find the nth term of: 20, 17, 14, 11, …

d = −3, so nth term starts with −3n.

When n = 1: −3(1) = −3, but the first term is 20, so add 23.

nth term = −3n + 23 (or equivalently 23 − 3n)

Worked Example 6

Is 150 a term in the sequence 7, 11, 15, 19, …?

nth term = 4n + 3

Set $4n + 3 = 150 \to 4n = 147 \to n = 36.75$4n+3=150→4n=147→n=36.75

Since n is not a whole number, 150 is not a term in this sequence.

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Quadratic Sequences [H]

A quadratic sequence has a second difference that is constant.

Finding the nth Term of a Quadratic Sequence

1. Find the first differences, then the second differences.
2. If the second difference is constant = 2a, then the $n^{2}$n2 coefficient is a.
3. Subtract an$^{2}$2 from each term to get the remaining linear sequence.
4. Find the nth term of that linear sequence.

Worked Example 7

Find the nth term of: 3, 9, 19, 33, 51, …

First differences: 6, 10, 14, 18

Second differences: 4, 4, 4 (constant$) \to 2a = 4$)→2a=4, so a = 2

The $n^{2}$n2 coefficient is 2, so subtract $2n^{2}$2n2 from each term:

n	1	2	3	4	5
Term	3	9	19	33	51
$2n^{2}$2n2	2	8	18	32	50
Term $- 2n^{2}$−2n2	1	1	1	1	1

The remaining sequence is 1, 1, 1, 1, $1 \to$1→ this is just the constant 1.

$nth term = 2n^{2} + 1$nthterm=2n2+1

Worked Example 8

Find the nth term of: 2, 9, 20, 35, 54, …

First differences: 7, 11, 15, 19

Second differences: 4, 4, $4 \to 2a = 4$4→2a=4, a = 2

Subtract $2n^{2}$2n2: 2 − 2 = 0, 9 − 8 = 1, 20 − 18 = 2, 35 − 32 = 3, 54 − 50 = 4

Remaining: 0, 1, 2, 3, $4 \to nth term = n - 1$4→nthterm=n−1

$nth term = 2n^{2} + n - 1$nthterm=2n2+n−1

Check: n = 3: 2(9) + 3 − 1 = 18 + 3 − 1 = 20 ✓

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Geometric Sequences [H]

Each term is found by multiplying the previous term by a constant common ratio r.

nth term = ar^{n-1}

where a = first term, r = common ratio.

Worked Example 9

Find the 8th term of the geometric sequence 3, 6, 12, 24, …

a = 3, r = 2

$8th term = 3 \times 2^{7} = 3 \times 128 =$8thterm=3×27=3×128= 384

Worked Example 10

A geometric sequence has first term 500 and common ratio 0.8. Find the 6th term.

$6th term = 500 \times 0.8^{5} = 500 \times 0.32768 =$6thterm=500×0.85=500×0.32768= 163.84

Worked Example 11

The 3rd term of a geometric sequence is 12 and the 6th term is 96. Find the common ratio and the first term.

$ar^{2} = 12$ar2=12 ... (1) $ar^{5} = 96$ar5=96 ... (2)

Divide (2) by (1): $r^{3} = 8 \to r = 2$r3=8→r=2

From (1): $a(4) = 12 \to a = 3$a(4)=12→a=3

First term = 3, common ratio = 2

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Special Sequences

Sequence	Terms	nth term
Square numbers	1, 4, 9, 16, 25, …	$n^{2}$n2
Cube numbers	1, 8, 27, 64, 125, …	$n^{3}$n3
Triangular numbers	1, 3, 6, 10, 15, …	n(n + 1)/2
Fibonacci	1, 1, 2, 3, 5, 8, 13, …	Each term = sum of two previous
Powers of 2	2, 4, 8, 16, 32, …	$2^{n}$2n

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Common Mistakes and Misconceptions