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Sequences and induction, grade 11 (5 units)

Sequences of numbers with a pattern, and a new way of proving – mathematical induction – for claims about all the natural numbers.Subjects

The course builds on the algebra of grade 10: equations and inequalities, systems of equations, the laws of powers and factorising. The topics, in order:

  • a general sequence: a formula by place, a recursion rule, a graph of points, increasing, decreasing and periodic sequences;
  • the arithmetic sequence: the general term, the first term and the difference, the sum, operations on the terms, sub-sequences and problems;
  • the geometric sequence: the general term and the ratio, the geometric mean, the sum, operations on the terms and sub-sequences, the infinite geometric sequence and problems of growth and decay;
  • mathematical induction: the principle, sum formulas, a recursion rule and a formula by place that agree, divisibility, and mistakes in proofs.

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 a formula by placeLearningA sequence as a function of the place: a table, a formula and a graph of points; a formula from a picture, two ways; increasing, decreasing and alternating sequences; the place of a term.
  2. A recursion ruleLearningA term from the term before it: a recursion rule and a starting condition; from a formula by place to a recursion rule; guessing a formula from the terms; a periodic sequence; a rule using two terms.
  3. Practice: sequencesPracticeA formula by place and a recursion rule, the place of a term, increasing and decreasing sequences, a periodic sequence and a sequence from a picture.
  4. The arithmetic sequenceLearningA constant difference: the recursion rule and the general term; a graph of points on a line; each term is the mean of its neighbours; the place of a term and the number of terms; a proof and a counterexample.
  5. The first term and the differenceLearningTwo facts – two equations in $a_1$ and $d$; a system with one quadratic equation; three neighbouring terms $a-d$, $a$, $a+d$.
  6. The sum of an arithmetic sequenceLearningThe sum twice, from the start and from the end; $S_n$ in two forms; stairs that make a rectangle; the partial sums on a parabola; the number of terms from the sum; the term from the sum.
  7. Operations on an arithmetic sequence and sub-sequencesLearningWhich operations on the terms keep a sequence arithmetic, and what they do to the graph; the terms in the even and the odd places; a sequence made of two arithmetic sequences.
  8. Arithmetic sequence problemsLearningA cinema, a salary that rises every month, two hikers walking towards each other, stairs and a pile of logs.
  9. Practice: the arithmetic sequencePracticeA term, a place and the number of terms, the first term and the difference, a sum, the term from the sum, operations on the terms, sub-sequences and a word problem.
  10. The geometric sequenceLearningA constant ratio: the recursion rule and the general term; the laws of powers with whole exponents, negative ones too; increasing, decreasing and alternating sequences; the place of a term by trial.
  11. The geometric mean, the first term and the ratioLearningA term's square is the product of its neighbours; the ratio from two terms, also when there are two sequences; a system in $a_1$ and $q$; three neighbouring terms $\frac{b}{q}$, $b$, $bq$.
  12. The sum of a geometric sequenceLearningThe sum times $q$ minus the sum; $S_n$ and the number of terms; alternating signs; how many terms it takes to pass a number; geometric against arithmetic growth.
  13. Operations on a geometric sequence and sub-sequencesLearningWhich operations on the terms keep a sequence geometric; from an arithmetic sequence to a geometric one by powers; the terms in the even places; a sequence made of two geometric sequences.
  14. The infinite geometric sequenceLearning$q^n$ tends to 0 when $|q|<1$; a convergent sum $S=\frac{a_1}{1-q}$; a repeating decimal as an infinite sum; for which values of $x$ there is a sum; squares inside squares.
  15. Growth and decay problemsLearningCompound interest, a salary rising by a percentage against one rising by shekels, bacteria, the chessboard legend, a bouncing ball and the Cantor set.
  16. Practice: the geometric sequencePracticeA term and a place, the ratio from two terms, a sum, sub-sequences, an infinite sequence, a repeating decimal and a decay problem.
  17. The principle of inductionLearningExamples are not a proof; the base case, the inductive step and the conclusion; a proof that the sum of the odd numbers is a square; a hypothesis against a claim; tiling boards with L pieces.
  18. Proving sum formulasLearningThe sum of the natural numbers, the sum of the squares and a geometric sum by induction; in the step – add one term, put in the hypothesis and take out a common factor.
  19. Induction and sequencesLearningA recursion rule and a formula by place that define the same sequence; guessing a formula and proving it; the general term of an arithmetic sequence; triangular and pentagonal numbers; lines that divide the plane.
  20. DivisibilityLearning$7^n-1$ is divisible by 6, two ways; in the step – write the expression for $k+1$ with the expression for $k$; $n^3+2n$ is divisible by 3.
  21. Mistakes in induction proofsLearningA step without a base, a base without a step, assuming what is being proved, and a claim checked only on examples: how to tell a proof that is no proof.
  22. Practice: mathematical inductionPracticeSum formulas, a recursion rule and a formula by place, divisibility and finding the mistake in a proof.
  23. Review: sequences and inductionPracticeQuestions from the whole course: a general sequence, arithmetic and geometric sequences, the infinite geometric sequence, a word problem and proofs by induction.
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