8If the values in this column are equal, then Tn = an3 + bn2 + cn + d If neither of the difference columns, d1 or d2 have equal values, proceed to the third difference column (d3)If the values in this column are equal, thenTn = an3 + bn2 + cn + d
9Exercise: Find the first 5 general terms of a third degree sequence and show d1, d2 and d3Tn = an3 + bn2 + cn + dT1 = a+b+c+dT2 = 8a+4b+2c+dT3 = 27a+9b+3c+dT4 = 64a+16b+4c+dT5 = 125a+25b+5c+d7a+3b+c12a+2b6a19a+5b+c18a+2b37a+7b+c6a24a+2b61a+9b+c
28n P 1 2 4 3 10 20 Find the general rule n=1 n=2 n=3 n=4 12431020n=1n=2n=3n=4In the figures above,n = the no. of units on each sideP = total no. of “point-up” triangles of all sizesFind the general rule
32Tn = 2n2 -2n + 1 1 5 4 13 8 4 25 12 2nd differences are common ∴ Tn = an2 + bn + c1513254812a + b + c = 1c = 12a = 4a = 243a + b = 4b = -2Tn = 2n2 -2n + 1
33Sequences other than APs and GPs Model: Write down the first 6 terms of the sequence in which t1=5 and tn+1=tn+3.tn+1=tn+3t2 = t = 5+3 = 8t3 = t = 8+3 = 11t4 = t = 11+3 = 14t5 = t = 14+3 = 17t6 = t = 17+3 = 20First 6 terms are 5, 8, 11, 14, 17, 20
34The Fibonacci Sequence The Fibonacci sequence was derived by Leonardo of Pisa who used the name Fibonacci for his published writings. The question that Leonardo posed that led to the development of the Fibonacci sequence was this one: How many pairs of rabbits can be produced from a single pair in one year if it is assumed that every month each pair begets a new pair which from the second month becomes productive?
38Fnr = Fn Fn-11231.551.666…81.6131.62521341.619…551.617…891.618…Use the statistical graphing capability of your calculator to produce a scatter graph of r against nThe approximate value of the limit of r is known as (phi) which is written as the surd
39 has some interesting properties. e.g. Consider what happens when you raise to increasing powers:Note: + 2 = 32 + 3 = 4Note also1 + = 2This means that1, , 2, 3, 4 form a Fibonacci sequence
40Do these numbers look familiar ? … and these also form a recursive function11
47Proof by Induction Steps: Prove true for n=1 State the proposition for n=kAssume the truth of the proposition for n=k and show that it is true for n=k+1If true for n=1, then it must be true for n=2If true for n=2, then it must be true for n=3etc
48Model: Prove that the sum of the first n squares, Sn= 12 + 22 + 32 +… Model: Prove that the sum of the first n squares, Sn= ….+n2 is given by Sn =Proof S1 = 12 = 1 Also S1 = = ∴ true for n=1 Assume true for n=k i.e. Sk= Now Sk+1 = Sk + (k+1)2 = + (k+1)2
49Model: Prove that the sum of the first n squares, Sn= 12 + 22 + 32 +… Model: Prove that the sum of the first n squares, Sn= ….+n2 is given by Sn == + (k+1)2 = + k2 + 2k + 1 Nowi.e. Rule is true for Sk+1∴ If true for n=1 then true for n=2If true for n=2 then true for n=3etcSk+1
50Exercise Use the method of proof by induction to prove: 1. the sum of the first n terms of a GP with first term, a, and common ratio, r,is given by2. the sum of the first n cubes, Sn = ….+n3 is given by3. the sum of the first n fourth powers, Sn = …+n4 is given by
52Model : Show that n(n+1)(n+2) is divisible by 6 When n=1, n(n+1)(n+2) = 1x2x3 = 6 ∴ true for n=1Assume true for n=ki.e. k(k+1)(k+2) = 6a (for some integer “a”)When n= k+1Tk+1 = (k+1)(k+1+1)(k+1+2)= (k+1)(k+2)(k+3)= k(k+1)(k+2) + 3(k+1)(k+2)= 6a + 3(k+1)(k+2) which must be a multiple of 6 since (k+1)(k+2) must be even= 6a + 6b for some integer “b”= 6(a+b)∴ true for n=k+1etc
54Hailstone sequences (p183) Sequences such as 5, 16, 8, 4, 2, 1, 4, 2, 1, …and 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, …are known as hailstone sequences because they bounce around before coming to rest.Hailstone sequences are generated as follows:Start with any positive integer n (Everybody choose one)If n is even, divide by 2 to get the next termIf n is odd, multiply by 3 and add 1 to get the next termRepeat this process with successive terms (Everybody try this)One of mathematics’ unsolved problems is to prove that every starting value will generate a sequence that eventually settles to 4, 2, 1, 4, 2, 1, …Could there possibly be a sequence that doesn’t settle down to this cycle?
56Think of a two digit number Think of a two digit number. Add together these 2 digits and subtract this sum from your original number. When you have the final number look it up on the chart below:I will now tell you the symbol associated with your number
57Think of a two digit number Think of a two digit number. Add together these 2 digits and subtract this sum from your original number. When you have the final number look it up on the chart below:I will now tell you the symbol associated with your number
58Methods of ProofProof by obviousness The proof is so clear that it need not be mentionedProof by general agreement All those in favourProof by imagination We’ll pretend that it’s true…Proof by necessity It had better be true or the entire structure of mathematics would crumble to the ground.Proof by plausibility It sounds good, so it must be true.Proof by intimidation Don’t be stupid. Of course it’s true!
59Proof by lack of sufficient Because of the time constraint, I’ll time leave the proof to you.Proof by postponement Because the proof of this is so long,it is given in the appendix.Proof by accident Hey! What have we got here?Proof by insignificance Who cares anyway?Proof by profanity (Example censored)Proof by definition We define it to be trueProof by lost reference I know I saw it somewhereProof by calculus This proof requires calculus, so we’ll skip it
60Proof by lack of interest Does anyone really want to see this? Proof by illegibilityProof by divine word And the Lord said, “Let it be true” and it was trueProof by intuition I just have this gut feeling
61THE ANSWER IS 22 Write down a three digit number 452454252542524242Write down a three digit numberWrite down as many 2 digit combinations of these 3 digits as possible and add these numbersAdd the digits of the original number4+5+2 = 11Divide the previous total by this numberProve thisTHE ANSWER IS 22
62381119304979128207335Write down any 10 numbers that have a Fibonacci type sequence i.e. tn + tn+1 = tn+2Add these numbersLet me see your numbers and I will quickly tell you what they add up toProve thisThe answer is 11 x t7
64Think of a 3 digit number Reverse the digits and take the smaller from the larger Call this number x Reverse the digits of x to give you another 3 digit number. Call this number y. Add x and y Your answer is 1089Prove thisx y 1089