Sequences
Term 1
Arithmetic · Geometric · Shifted patterns · Recursive notation
Learning Objectives
PrintRepresenting Sequences
- 1. Represent a sequence using pictures, tables of values, graphs and algebra.
- 2. Describe a sequence using a term-to-term rule (in words) and a position-to-term rule (using $n$).
- 3. Identify whether a sequence is arithmetic, geometric, quadratic or neither.
- 4. Use the general rule of a sequence to predict any $n$th term.
- 5. Test whether a given number is a term of a sequence by solving an equation.
Arithmetic and Geometric Sequences
- 6. Determine the general rule $u_n = u_1 + (n-1)d$ for an arithmetic sequence.
- 7. Determine the general rule $u_n = u_1 \cdot r^{n-1}$ for a geometric sequence.
- 8. Work backwards from given terms to find $u_1$, $d$, $r$ or $n$ (e.g. given $u_5 = 2$ and $u_8 = 8$, find $r$).
- 9. Use properties of sequences to solve for unknowns (e.g. consecutive arithmetic terms $k$, $2k+2$, $4k+4$; consecutive geometric terms $2$, $k$, $10$).
- 10. Write a recursive rule for an arithmetic sequence using $u_{n+1} = u_n + d$.
- 11. Write a recursive rule for a geometric sequence using $u_{n+1} = u_n \cdot r$.
Shifted Patterns and Series (Extended)
- 12. Determine the general rule from a shifted square or cube sequence (e.g. one less than the square numbers gives $u_n = n^2 - 1$).
- 13. Calculate the sum of the first $n$ terms of an arithmetic series.
- 14. Calculate the sum of the first $n$ terms of a geometric series.
- 15. Discuss the behaviour of an infinite geometric series and identify when it converges.
Assessments
Sequences end-of-unit test
Criterion A
60 min
Written test on arithmetic and geometric sequences, finding the nth term and the sum of a series, and using recursive notation.
Frogs investigation
Criterion B
60 min
Students explore the classic Frogs puzzle (swapping frogs of two colours across a row of lily pads), record the minimum number of moves needed for different numbers of frogs, find patterns in the data, conjecture a general formula, and verify it for a larger case.
Review Pack
Open review hub
Fluency Pack A — 40 questions
Fluency Pack B — 40 questions
Problem-solving — 12 unfamiliar tasks
Worked solutions (Pack A + B + problems)