1. Proof by induction
There are different ways to prove in Maths, induction uses a ‘domino effect’:
Step 1: test that an algebraic statement is true for one case, when n = 1
Step 2: assume it is true for a general case, when n = k
Step 3: show that the statement holds for the next case, n = k + 1
Step 4: conclude that as the first case is true, and that subsequent cases are true, the statement must be true for all cases
This technique is applied to 4 different types of statement in FP1:
Proving the summation of a series
Proving divisibility
Eg prove that is divisible by 4
Eg prove that
Proving a general term for a recurrence relation
Proving a matrix power
Eg Given that
Eg prove that
prove that

2. WB35(a) Prove by induction that, for any positive integer n,
LHS = RHS, true for n = 1
Assume that
using assumption
as required
true for n = k + 1 if true for n = k.
True for n = 1, true for all n.

3. WB36(a) Prove by induction that,
LHS = RHS, true for n = 1
Assume that
using assumption
Now try Ex 6A
as required
true for n = k + 1 if true for n = k.
True for n = 1, true for all n.

4. Ex 6A Q7 Q6 Q9

5. Recurrence relationship
WB31 A sequence of numbers is defined by
Prove by induction that, for n  ℤ,
nth term rule
nth term rule
as given
nth term rule
true for n = 1
Recurrence relationship
Assume that
by recurrence relationship
using assumption
expanding
as required
true for n = k + 1 if true for n = k.
True for n = 1, true for all n.

6. Recurrence relationship
WB32 A sequence of numbers u1, u2, u3, u4, . . ., is defined by
Prove by induction that, for n  ℤ+,
nth term rule
Recurrence relationship
nth term rule
true for n = 1
Assume that
by recurrence relationship
using assumption
Now try Ex 6C
as required
true for n = k + 1 if true for n = k.
True for n = 1, true for all n.

7. WB33 Prove by induction, that for n  ℤ+,
true for n = 1
Assume that
true for n = k + 1 if true for n = k.
True for n = 1, true for all n.
Now try Ex 6D

8. Ex 6D Q2 Q4 Q3

9. (b) is divisible by 12
true for n = 1
Assume that for some A  ℤ
To prove divisibility, use the fact that: so
true for n = k + 1 if true for n = k.
True for n = 1, true for all n.

10. WB34
(a) Show that
b) Hence, or otherwise, prove by induction that, for n  ℤ+, f(n) is divisible by 8.
true for n = 1
Assume that for some A  ℤ
Now try Ex 6B
using (a)
true for n = k + 1 if true for n = k.
True for n = 1, true for all n.

11. The divisibility method must be adapted at times to achieve the required proof:
By using prove by induction that, for n  ℤ+, f(n) is divisible by 9. Eg
Assume that for some A  ℤ
Why consider instead of the usual ?
This is not obviously divisible by 9 - the standard method fails

12. If the method is adapted, the proof may require some extra explanationEg
By using
prove by induction
that, for n  ℤ+, f(n) is divisible by 5.
Assume that for some A  ℤ
using earlier result
This tells us that
But as 2 and 5 are coprime, f(k+1) must be divisible by 5
for some B  ℤ
This statement validates the proof and must be included
Coprime numbers share no factors

13. The general term given a recurrence relation
Proving:
Example
Method
Matrix powers
Summation of a series
The general term given a recurrence relation If
given
given
then
that
Divisibility
is divisible by 4