Algebra - Fibonacci, Arithmetic and Geometric Sequences, Study notes of Mathematics

Download Algebra - Fibonacci, Arithmetic and Geometric Sequences and more Study notes Mathematics in PDF only on Docsity! Page 1 of 7 Algebra Algebra - Sequences Starter Today we will talk about: 1. Sequences 2. Fibonacci Sequences 3. Arithmetic Sequences 4. Geometri Sequences Sequences A Sequence is simply a set of numbers (or objects) in an order. Like 2, 4, 6, 8... Is a sequence of even numbers. 1, 3, 5, 7... is a Sequence of odd numbers. You can have many sequences like Fibonacci, Triangle, Arithmetic and Geometric etc. We will talk about triangle numbers when we do trigonometry. Fibonacci Sequence This is probably the most famous sequence in mathematics. Fibonacci sequences are sequences where each term is the sum of the previous two terms. In a Fibonacci sequence, the first two numbers must be chosen before starting the sequence. The most common sequence begins with 0 and 1. This specific sequence of numbers generated by 0 and 1 are known as the “Fibonacci numbers”. Since the first two terms are 0 and 1, the third term will be the first term and the second term added together (0 + 1) which gives 1. So the third term is also 1. The fourth term will be the second and third terms added together (1 + 1) which gives 2. The fifth term will be the third and fourth added together (1 + 2) which gives 3. And so on... 0 1 1 2 3 5 8 13 O+1 1+1 1+2 2+3 3+5 5+8 Page 2 of 7 Algebra Arithmetic (Linear) Sequences A Linear Sequence is one where the numbers go up (or down) by the same amount each time - you add each term by a common difference to get to the next term. Eg: 1, 4, 7, 10, 13, 16, 19... : where the first term is 1 and then we add 3 to get the next term. 10, 5, 0, -5, -10... : where the first term is 10 and then we subtract 5 (or add -5) to get the next term. You should be able to recognise and continue a Linear Sequence. You should also be able to find a formula for the nt term of a Linear Sequence in terms of n. This formula will be in the form nt term =dn+a where o dis the common difference o ais the zeroth term of the sequence Example 1: Find the nth term of the sequence 4, 6, 8, 10, 12... 1. Find the common difference by subtracting two consecutive numbers like (6-4) = (8-6) = (10-8)...= 2. d (common difference) = 2 2. Find the zeroth (Ot) term by subtracting the first by the common difference: 4-2 =2.a=2 3. So equation = 2n+ 2 4. Lets check our equation: 1. To check the first term, substitute 1 in our equation: (2x1) +2 =4 2. To check the second term, substitute 2 in our equation: (2x2) + 2 =6 3. To check the third term, substitute 3 in our equation: (2x3) + 2 =8 Hence our equation is correct. Example 2: Find the nth term of the sequence 11, 7, 3, -1, -5... 1. Find the common difference by subtracting two consecutive numbers like (7-11) = (8-7) = (-1-3)...= -4. d (common difference) = -4 2. Find the zeroth (Ot) term by subtracting the first by the common difference: 11 - (-4)=11+4=15.a=15 3. So equation = -4n + 15 4. Lets check our equation: Page 5 of 7 Algebra 1. Find the common ratio by dividing two consecutive numbers like (2)2+2) = (4+2J2)...= J2. r (common ratio) = /2 3. Find the zeroth (0') term by dividing the first by the common ratio: ¥2+/2=1.a=1 4. So equation = 1 x (/2)n 5. Lets check our equation: 1. To check the first term, substitute 1 in our equation: 1 x (/2)1 = 2 2. To check the second term, substitute 2 in our equation: 1 x (J2)2 =2 3. To check the third term, substitute 3 in our equation: 1 x (/2)3 = 2/2 Hence our equation is correct. Now try: Find the nth term of the following sequences: 1. 27,93, 9, 3/3... 2. J9, 3/2, 6, /72... Quadratic Equations Unlike in a linear sequence, in a quadratic sequence the differences between the terms (the first differences) are not constant. However, the differences between the differences (the second differences) are constant. You should also be able to find a formula for the nt term of a Linear Sequence in terms of n. This formula will be in the form: n* term = an? + bn +c where o ais the (second difference / 2) o (bn-+c) is the equation of the sequence after subtraction Example: Find the equation of the sequence 2, 3, 6, 11, 18, ... Ans: Start by making a table: Term 1 2 3 4 5 Quadratic 2 3 6 an 18 Sequence First 3-25 6- Difference 1 Second 3-1= 5-3= 7-5= Difference 2 2 2 Page 6 of 7 Algebra a = second difference /2=2/2=1 Now subtract 1n2 from the quadratic sequence: Quadratic 2 3 6 an 18 Sequence 1n2 1 4 9 16 25 Quadratic 1 -1 3 5 -7 Sequence - 1n2 Now find the equation of the new sequence. The equation is -2n + 3. Combine both the parts to get the final equation: 1n?-2n+3 Lets verify: ni) =(1x 1)-(2x1)+3=2 n(2) = (1 x 4)-(2x2)+3=3 n(3) = (1 x 9) - (2x 3)+3=6 Hence our equation is correct. Now try: Find the nth term of the following sequences: 1. 5,11,21,35... 2. 9,13, 19, 27... 3. 2,10, 24, 44... 4. 19,15,9,1... Term-to-term Progression So far we’ve been dealing with position-to-term progression of the form an = f(n). Progression of this firm lets us find the term value only from the position. In term-to-term progression you can only find a term if the term before it is known. A term-to-term rule gives the (n+1)t» term in terms of the nt term * i€ ans is given in terms of an « Ifa term is known, the next one can be worked out Page 7 of 7 Algebra This formula will be in the form: An+1 = an +d TERM-TO-TERM RULE where o ais the n term. n 4 2 3 4 °5 o dis the common a 4 5 67 8 difference. THE LINK BETWEEN ONE TERM (a,) AND THE NEXT (ay44) IS TO ADD ONE Now try: Find the term-to-term equation of SO THE TERM-TO-TERM RULE IS a,.,=a,+4 the following sequences: 1. 18,16, 14, 12, 10... 2. 3,6, 9,15, 21... 3. 9 (a) Find the first 5 terms of the following sequences oO (i) n" term = 20 — 2n(ii) n" term = 2n? — 3 1 iii) n term = — (iii) 5 n (b) Find the 2"4, 3rd and 4th terms in each of these sequences (i) ay4, =a,+6 with a, =3 (ii) 4,4) = 3a,,-1 with a,=2 (iii) Gy4.=Ay4, +4, with a,=4anda,=6