Covers the various sections relating to Mathematical Induction within the Advanced Mathematics programme.
Includes notes from the textbook, as well as additional class, video and research information, diagrams and practice questions.
Applicable to all IEB Grade 12s.
Mathematical Induction
= helps to prove statements
Steps:
1. Let S(n) be a statement involving the natural number, n
2. Prove that statement is true for n = 1
3. Assume the statement is true for n = k where k € N
4. Prove true for n = k + 1
5. By the principle of MI, the statement is true A n € N
Two types:
- Divisibility
- Summing
Example of divisibility:
Prove 5n - 1 is divisible by 4 for n ∈ N
Step 1: Prove that statement is true for n = 1
51 - 1 = 4
4
. =1
4
∴ The statement is true for n = 1
Step 2: Assume the statement is true for n = k where k ∈ N
5k - 1 is divisible by 4
5𝑘−1
. 4
= p where p ∈ N
5k - 1 = 4p
. 5k = 4p + 1
Step 3: Prove true for n = k + 1
5k + 1 – 1
= 5k.51 – 1
= (4p + 1).5 – 1
= 20p + 5 – 1
= 20p + 4
= 4 (5p + 1) which is divisible by 4.
Step 4: By the principle of MI, the statement is true A n ∈ N
Example of a summing:
Prove that 2 + 4 + 6 + ... + 2n = n (n + 1) for n ∈N
LHS:
T1 = 2
T2 = 4
T3 = 6
T4 = 2n
RHS:
Sn = n(n + 1)
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