Proof by induction There are different ways to

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 FP 1: Proving divisibility Eg prove that is divisible by 4 Proving a general term for a recurrence relation Eg Given that prove that Proving the summation of a series Eg prove that Proving a matrix power Eg prove that

WB 35(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.

WB 36(a) Prove by induction that, true for n = 1 LHS = RHS, Assume that using assumption Now try Ex 6 A as required true for n = k + 1 if true for n = k. True for n = 1, true for all n.

Ex 6 A Q 7 Q 6 Q 9

Recurrence relationship WB 31 A sequence of numbers is defined by Prove by induction that, for n nth term rule ℤ, 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.

Recurrence relationship WB 32 A sequence of numbers u 1, u 2, u 3, u 4, . . . , is defined by Prove by induction that, for n ℤ+, nth term rule Recurrence relationship true for n = 1 nth term rule Assume that by recurrence relationship using assumption Now try Ex 6 C as required true for n = k + 1 if true for n = k. True for n = 1, true for all n.

WB 33 Prove by induction, that for n ℤ+, (a) true for n = 1 Assume that true for n = k + 1 if true for n = k. True for n = 1, Now try Ex 6 D true for all n.

Ex 6 D Q 2 Q 3 Q 4

(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.

WB 34 (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 using (a) true for n = k + 1 if true for n = k. True for n = 1, Now try Ex 6 B true for all n.

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 Why consider for some A ℤ instead of the usual ? This is not obviously divisible by 9 - the standard method fails

If the method is adapted, the proof may require some extra explanation Eg 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 Coprime numbers share no factors for some B ℤ This statement validates the proof and must be included

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