Proof by Induction (Edexcel International A Level Further Maths)

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Expertise

Maths

Proof by induction is a way of proving a result is true for a set of integers by showing that if it is true for one integer then it is true for the next integer
It can be thought of as falling dominoes:
- Assume one domino falls
  - The assumption step
- Show that if this domino falls, the next domino falls
  - The inductive step
If you want all dominoes to fall (from the beginning) then
- Show also that the first domino falls
  - The basic step

STEP 1
The basic step: Show the result is true for the base case
- This is normally $n = 1$ or $n = 0$

STEP 2
The assumption step: Assume the result is true for $n = k$ where $k$ is some integer
- There is nothing to do for this step apart from writing down the assumption

STEP 3
The inductive step: Use the assumption to show the result is true for $n = k + 1$
- This involves investigating $n = k + 1$ and bringing in the $n = k$ assumption

STEP 4
The conclusion step: Explain in words how the above steps make the result true for all integers, using the following two sentences:
- "If it is true for $n = k$ , then it is true for $n = k + 1$ ."
- "As it is true for $n = 1$ , the statement is true for all $n \in ℤ^{+}$ ."

Learn the exact wording from the conclusion above! You can lose the final mark if your conclusion does not make sense.

to absolutely everything:

the (exam) results speak for themselves:

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