Level 5 · Mathematics M5.8

Proof, sequences and series

Learn to prove things for certain, and work with sequences and their sums.

Planned. This book is not written yet. Below is its plan, which a teacher can already use.

You are ready for this book if

You can find the nth term of 7, 11, 15, 19.

The plan

  1. M5.8.1 What a proof is
    • Prove that odd × odd is always odd using algebra
    • Disprove a claim with one counterexample
    • Prove a result by checking every case
  2. M5.8.2 Proof by contradiction
    • Prove that √2 is irrational
    • Prove that there are infinitely many primes
  3. M5.8.3 Sequences
    • Write the terms of a sequence from an nth term rule
    • Build a sequence from a rule that uses the term before
    • Say if a sequence goes up, goes down or repeats
  4. M5.8.4 Sigma notation
    • Read and write sums using Σ
    • Find the sum of 2r + 1 for r = 1 to 10
  5. M5.8.5 Arithmetic series
    • Add 1 + 2 + 3 up to 100 by pairing the terms
    • Find the sum of the first n terms of an arithmetic series
  6. M5.8.6 Geometric series
    • Find the nth term and the sum of a geometric series
    • Find the sum to infinity when it exists
  7. M5.8.7 Proof by induction
    • Prove a formula for a sum by induction
    • Prove a result about dividing by induction
  8. M5.8.8 Limits and series that settle
    • Find the limit of a sequence like (n + 1)/n
    • Show that 1 + 1/2 + 1/3 + 1/4 ... grows without limit

For parents and teachers

Level 5 fits learners who can already: read a textbook. use functions and graphs. In many schools that is about 17 to 20, but start wherever the learner is.

Strand: Algebra. Written and drawn by attnom. Free to read, print and share.