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Corbett MathsArithmetic sequences — nth term
Corbett MathsComposite and inverse functions
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Key facts & methods
Arithmetic sequences
- nth term = a + (n−1)d where a = first term, d = common difference.
- E.g. sequence 3, 7, 11, 15... → a=3, d=4. nth term = 3 + (n−1)×4 = 4n − 1.
- Sum of n terms: Sₙ = n/2 × (2a + (n−1)d) = n/2 × (first + last).
- Geometric sequences: each term multiplied by a constant ratio r. nth term = ar^(n−1).
- Quadratic sequences: second differences are constant. nth term includes an n² term.
Functions — notation and operations
- f(x) = ... means "f is a function of x". f(3) means substitute x=3.
- Composite function: fg(x) = f(g(x)). Apply g first, then f to the result.
- Inverse function f⁻¹(x): swap x and y, then rearrange for y.
- E.g. f(x) = 2x+3. Let y = 2x+3, swap: x = 2y+3, rearrange: y = (x−3)/2. So f⁻¹(x) = (x−3)/2.
- Domain = set of valid inputs. Range = set of possible outputs.
- ff⁻¹(x) = x always (applying a function then its inverse returns the original value).
Exam questions — 5 questions · 12 marks · Edexcel 4MA1 style
⚡ Show ALL working — the AI awards method marks just like a real examiner. Partial credit for correct working even if the final answer is wrong.
1 marknth term4MA1 style
The nth term of a sequence is 3n − 2. What is the 10th term?
3 marksArithmetic sequence problem4MA1 style ✏️ No calculator
An arithmetic sequence has first term 7 and common difference 4. (a) Write down the nth term. (b) Find the sum of the first 20 terms. (3 marks)
Hint: (a) nth term = a + (n−1)d. Simplify. (b) Sₙ = n/2 × (2a + (n−1)d). Substitute n=20, a=7, d=4.
+30 XP
3 marksComposite and inverse functionsGrade 7 Booklet / 4MA1 style ✏️ No calculator
f(x) = 3x − 1 and g(x) = x² + 2. (a) Find fg(2). (b) Find f⁻¹(x). (c) Solve f⁻¹(x) = g(1). (4 marks)
Hint: (a) fg means apply g first, then f. (b) Swap x and y, rearrange for y. (c) Find g(1) first, then set f⁻¹(x) equal to that value and solve.
+30 XP
1 markQuadratic sequence4MA1 style
A sequence begins: 2, 5, 10, 17, 26... What is the nth term?
A 3n − 1
B n² + 1
C 2n + 1
D n² + n
Module complete! 🎉
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