Skip to content

Sequences, grade 12 (4 units)

Sequences of numbers with a pattern: seats in the rows of a hall, payments, views of a video and a bouncing ball.Subjects

The course builds on the algebra of grades 7–9: equations, a system of two equations, the quadratic formula and the laws of powers. The topics, in order:

  • sequences: a formula by place, a recursion rule and a discrete graph;
  • the arithmetic sequence: the general term, a term’s place and the number of terms, the arithmetic mean, the sum, proving that a sequence is arithmetic, sub-sequences and problems;
  • the geometric sequence: the general term, the ratio and the geometric mean, the sum, proving that a sequence is geometric, the infinite geometric sequence and problems;
  • problems combining an arithmetic and a geometric sequence.

There are two kinds of lessons:

  • learning lessons – an explanation, worked examples and a few questions to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. A sequence and its general termLearningA sequence, a term and its place; the notation $a_n$; a formula by place; a sequence in words, in a table and on a discrete graph.
  2. A recursion ruleLearningEach term from the term before it: a recursion rule and the first term; working out terms; a formula by place and a recursion rule for the same sequence, and from a formula to a rule.
  3. Practice: a formula by place and a recursion rulePracticeTerms from a formula and from a recursion rule, the place of a term, a sequence on a graph, and from a formula to a recursion rule.
  4. The arithmetic sequenceLearningA constant difference between neighbouring terms; the sequence in words, in a table, by a recursion rule and on a graph – points on a line; the general term; increasing, decreasing and constant sequences.
  5. A term's place and the number of termsLearningAn equation in $n$: in which place a term is, and a number that is not a term; how many terms a finite sequence has; how many terms are positive; from a recursion rule to a formula and back.
  6. The first term, the difference and the arithmetic meanLearningTwo terms given: a system of equations in $a_1$ and $d$; steps of $d$; each term is the arithmetic mean of its neighbours; three neighbouring terms.
  7. The sum of an arithmetic sequenceLearningAdding in pairs; two formulas for the sum $S_n$; seats in a hall; how many terms to add for a given sum.
  8. Proving that a sequence is arithmeticLearningA general proof: $a_{n+1}-a_n$ is the same for every $n$; a counterexample; from a recursion rule; from the sum to the term: $S_n-S_{n-1}=a_n$.
  9. Sub-sequences of an arithmetic sequenceLearningThe terms in the odd and the even places; how many terms a sub-sequence has; the seats in the even rows; a new sequence from an arithmetic one; the sum of the last terms.
  10. Word problems with arithmetic sequencesLearningMonthly savings, a training plan, a pile of pipes, a loan and shooting practice: what the terms are, what is given and what is asked.
  11. Practice: the arithmetic sequencePracticeA term, a place and the number of terms, the first term and the difference, a sum, a proof from $S_n$, the arithmetic mean, a hall with two wings and payments for a television.
  12. The geometric sequenceLearningA constant ratio between neighbouring terms; the sequence in words, in a table, by a recursion rule and on a graph – points on the graph of an exponential function; the general term; increasing, decreasing, and alternating signs.
  13. The ratio, the first term and the geometric meanLearningTwo terms given: dividing one equation by the other; an even power and two sequences; each term is the geometric mean of its neighbours; the place of a term – powers of the same base.
  14. The sum of a geometric sequenceLearningThe formula for the sum $S_n$ and where it comes from; grains of rice on a board; a negative ratio; sub-sequences of a geometric sequence.
  15. Proving that a sequence is geometricLearningA general proof: $\frac{a_{n+1}}{a_n}$ is the same for every $n$; a counterexample; from a recursion rule; from the sum to the term in a geometric sequence.
  16. The sum of an infinite geometric sequenceLearningSums that approach a number; the formula $S=\frac{a_1}{1-q}$ when $-1<q<1$; a negative ratio; from the data to the sum; midsegment triangles; a bouncing ball.
  17. Word problems with geometric sequencesLearningViewers of a video and discount coupons, a water tank filling up, a salary rising by a percentage, a car's value and a chain message.
  18. Practice: the geometric sequencePracticeA term, the ratio and the first term, a proof, a sum, the geometric mean, the infinite geometric sequence and an app's downloads.
  19. Arithmetic and geometric sequences togetherLearningThree numbers in an arithmetic sequence that become a geometric one; a run there and back; two savings plans; terms of an arithmetic sequence that form a geometric one.
  20. Review: sequencesPracticeQuestions from the whole course: a recursion rule, the arithmetic sequence – the number of terms, a sum and a system of equations, a proof from $S_n$, the geometric sequence and its sum, the infinite geometric sequence and both kinds together.
Last updated on