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This Higher GCSE scheme of work unit covers sequences in Year 10. It finds a linear nth term, then a picture-sequence rule, generates terms from a formula, finds a quadratic nth term, writes a geometric nth term, and finishes with a recurrence.
Students have to treat n as a position, not as the next jump, and they have to use ar^(n−1) rather than ar^n. Those two distinctions are what later iteration, proof and compound-growth items all use.
A student who has secured this unit:
Students should be secure with the following before beginning this unit.
n is the term’s place in the list. Writing n + 5 for 3, 8, 13, 18 describes the difference, not the position-to-term rule.
A constant first difference means linear. A constant second difference means quadratic. The n² coefficient is half of that second difference.
A geometric sequence multiplies by a constant r. Its nth term is ar^(n−1), so term 1 is a, not ar.
A recurrence gives u_(n+1) from u_n. It is not an nth term. You still need a starting value.
| Misconception | Teaching focus |
|---|---|
| Treating n as the next term, so 3, 8, 13, 18 is written as n + 5. | Ask for the 1st term from the proposed formula. If it is not 3, n has been used as a jump. |
| Using ar^n for a geometric sequence instead of ar^(n−1). | Substitute n = 1 into the proposed formula. The output has to be the first term a, not ar. |
| Stopping at the n² piece of a quadratic and ignoring the leftover linear sequence. | Take the quadratic part away from each term and find the nth term of the list that remains. |
| Mixing a recurrence with an nth term, so u_(n+1) = 2n + 1 is treated as the 2n + 1 sequence. | Ask whether the rule uses n or the previous term before any values are generated. |
| Taking the common ratio as the first difference of a geometric list. | Divide consecutive terms, rather than subtracting them, on the first geometric examples. |
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Students learn how to work out the nth term of a linear sequence.
As learning progresses they use the nth term to model picture sequences.
Students learn how to find and apply the nth term of a picture sequence.
Later, as learning progresses, they use the nth term to set up and solve equations.
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Students learn to generate sequences using the nth-term formula.
As learning progresses the sequences involve brackets and powers.
Students learn how to find the nth term of a quadratic sequence.
Later, as learning progresses, they find the nth term from the coordinates of a quadratic graph.
Students learn how to work out the nth term of a geometric progression in the form arn-1.
As learning progresses, they consider progressions with fraction and negative common ratios.
Students learn how to generate a recurrence sequence.
Later, as learning progresses, they work out the formula from the sequence.
Additional practice working out the nth term of arithmetic sequences.
The problem-solving activity challenges students to model a picture sequence using the nth term.
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Students gain additional practice working with geometric progressions.
The problem-solving activity challenges students to use a geometric progression to model a real-life problem..
nth term of a geometric sequence
Additional practice generating quadratic sequences and finding the nth term of a quadratic sequence.
Later, as learning progresses, students create a quadratic sequence to find the number of matches played in a league.
Students practise generating terms of linear and quadratic sequences, before extending their understanding to more complex non-linear sequences such as geometric, reciprocal and exponential sequences.
Students revise how to derive the nth term of an arithmetic sequence.
As learning progresses, they use the nth term to find any sequence term when given its position.
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Students revise how to calculate missing terms and the nth term of quadratic and geometric sequences.
Later, as learning progresses, they link sequences to algebraic proof and setting up equations.
nth term of a geometric sequence
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