Revision notes for Pure 1(9709/12)

Revision notes for Pure 1(9709/12) By WaqasSuleman A-Level Teacher Beaconhouse School System

Contents 1. Sequence and Series 2. Functions & Quadratics 3. Binomial theorem 4. Coordinate Geometry 5. Trigonometry 6. Circular Measure 7. Vectors 8. Differentiation 9. Integration

Arithmetic Progressions If you have the sequence 2, 8, 14, 20, 26, then each term is 6 more than the previous term. This is an example of an arithmetic progression (AP) and the constant value that defines the difference between any two consecutive terms is called the common difference. If an arithmetic difference has a first term a and a common difference of d, then we can write a, (a + d), (a + 2d),... {a + (n-1) d} where the n th term = a + (n 1)d Sum of Arithmetic series The sum of an arithmetic series of n terms is found by making n/2 pairs each with the value of the sum of the first and last term. (Try this with the sum of the first 10 integers, by making 5 pairs of 11.) This gives us the formula: where a = first term and l = last term. As the last term is the n th term = a + (n 1)d we can rewrite this as: (Use the first formula if you know the first and last terms; use the second if you know the first term and the common difference.) Geometric Progressions If you have a sequence such as: 81, 27, 9, 3, 1, 1/3, 1/9,... then each term is one third of the term before. This can be written as 81, 81(1/3), 81(1/3) 2, 81(1/3) 3, 81(1/3) 4,... It is an example of a Geometric Progression (GP) where the each term is a multiple of the previous one. The multiplying factor is called the common ratio. So a GP with a first term a and a common ratio r with n terms, can be stated as a, ar, ar 2, ar 3, ar 4...ar n-1, where the n th term = ar n-1 Example:

In the sequence, 400, 200, 100, 50,... find the 8 th term. a = 400, r = 0.5 and so the 8 th term = 400 0.5 7 = 3.125 Note: To find which term has a certain value you will need to use logarithms. Example: In the sequence, 2, 6, 18, 54... which is the first term to exceed 1,000,000? a = 2, r = 3. 2 3 n-1 > 1,000,000 3 n-1 > 500000 (n 1) log 3 > log 500000 n > 12.94 Therefore: n = 13 Example: In the earlier sequence, 400, 200, 100, 50... which is the first term that is less than 1? 400 0.5 (n-1] < 1 0.5 (n-1) < 0.0025 (n-1) log 0.5 < log 0.0025 Therefore: n > 9, or n = 10 Note: The inequality sign changed because we divided by a negative (log 0.5 < 0) Sum of Geometric series The sum of the terms can be written in two ways.

where a = first term, r = common ratio and r 1. (use this formula when r < 1). Example: Evaluate, (Note: there are 9 terms.) The first term is when n = 2 (i.e 2.36 2 = 5.5696) Using the formula for the sum of a geometric progression gives: which is approximately 9300 (to 3 s.f.). Convergence The sum of an infinite series exists if: -1 < r < 1 or r < 1 This is because each successive term is getting smaller and so the series will tend towards a certain limit. This limit is found using the second of our two formulae: If r < 1 then as n, r n 0 and so:

Example: the series 1/3 + (1/3) 2 + (1/3) 3 + (1/3) 4 +... converges and its sum is 1 as n approaches. (A sequence such as n 3 has the first 6 terms as 1 + 8 + 27 + 64 + 125 + 216. As n approaches infinity, the sum also increases. Therefore, it is not convergent. This series is divergent. Every AP has a sum that approaches infinity as n increases, so every AP is divergent.) Example Find 1-1/2 + 1/4-1/8 +... 1-1/2 + 1/4-1/8 +... = 1 + (-1/2) + (-1/2) 2 + (-1/2) 3 +... This is a geometric progression where r = -½, so r < 1. Therefore this series converges to: Two final pieces of information that may be useful: Example: The 7 th term of a GP is 6, the 9 th is 1.5. The 8 th term is: (6 1.5) = 9 = 3 Here r = 0.5 and a = 384.

Homework Questions 1 Terms of a Sequences 1. Find the next 3 terms of the following sequences and state the rule to find the next term in each case a) 5, 9, 13, 17 b) 1, 3, 5, 7 c) 9, 13, 17, 21 d) -2, 6, 14, 22 e) 15, 22, 31, 42, 55 f) 4, 13, 26, 43, 64

g) 3, 7, 13, 21, 31 h) 5, 12, 21, 32, 45 i) 1, 2, 2, 4, 8 j) 1, 3, 6, 10, 15

Homework Questions 2 Using the Nth Term of A Sequences 1. Find the value of U 1, U 2, U 3 and U 20 a) U n = 3n b) U n = 7n 2 c) U n = 2n 2 d) U n = n 2 4 2. A sequence is generate according to the formula U n =an-b. Given that U 3 =7 and U 5 =13.find the value of a and b 3. Find the value of n for which U n =(3n-2) 2 has the given value of U n =100 4. A sequence is generated from the formula U n =pn 2 -q where p and q are constants. Given that U 1 =- 1 and U 3 =7, find the value of the constants p and q.

5. Find the value of n for which U n has the given value a) U n =4n-1 and U n =23 b) U n = 2n3 1 3 and U n = 5 c) U n = 5n + 6 and U n = 31

Homework Questions 3 Recursive Formula 1. Find the next 3 terms of the following sequences given both the first term and the recursive formula. a) U 1 = 5 U n + 1 = 3U n b) U 1 = 3 U n + 1 = 2U n c) U 1 = 2 U n + 1 = 3U n 4 d) U 1 = 16 U n + 1 = U n 4 2. By writing down the first 4 terms or otherwise, find the recursive formula that defines the following sequence. a) U n =2n-1

b) U n =3n-2 3. Find the next 4 terms of these recursively defined sequences a) U n+1 =U n -U n-1 when U 1 =6 and U 2 =2 b) U n+1 =3U n +2U n-1 when U 1 =1 and U 2 =-3 c) U n+1 =5U n -11 when U 1 =3 4. Write down the first 3 terms of the sequence defined by U n+1 =12-U n when U 1 =10

Homework Questions 4 General Term of an Arithmetic Sequence 1. Which of the following sequences are arithmetic? a) 7, 17, 27, 37 b) 12, 5, 0, -9, -17 c) 24, 15, 6, -3, -12 2. a) Find the 10 th term and b) Find the formula for the nth term a) 4, 7, 10, 13 b) -3, -1, 1, 3 c) 1, -4, -9, -13

3. Find the 20 th term, if the sequence begins a) 2, 6, 10, 14, 18 b) 5, -3, -11, -19 c) 21, 27.5, 34, 40.5, 47 4. Find the number of terms in the arithmetic sequence 4, 9, 14, 19 169

Homework Questions 5 Arithmetic Sequences 1. Find the number of terms in the following sequences if you are given the first few and the last term. a) 12, 25, 38,..155 b) 198, 192, 186, 180,.78 2. Find the first term of the sequence and the common difference if a) U 2 = 2 U 5 = 17 b) U 4 = 10 U 8 = 6 3. Find the 22 nd term and the nth term of the following sequences a) 5, 11, 17, 23.. b) 25, 21, 17, 13.

4. If the first term of an arithmetic sequence is 8 and the common difference is -5. what is the 22 nd term? 5. An arithmetic sequence has a first term of 15 and the 8 th term is 43. What are the first four terms of the sequence? 6. The first two terms of an arithmetic sequence are a+2b and 7b. Find the 3 rd term. 7. What is the common difference of the arithmetic sequence with a 6 th term of -56 and an 11 th term of 11?

Homework Questions 6 Partial sums of Arithmetic Sequences 1. Find the sum of the following series a) 17, 25, 33, 41 (25 terms) b) 15, 26, 37, 42.(15 terms) c) 143, 130, 117, 104.(22 terms) d) 96, 90.5, 85, 79.5..(21 terms) 2. Find the sum of the following arithmetic sequences if you are given the first and the last term a) 5, 19, 33, 47,..243

b) 271, 263, 255, 247,..95 c) 78, 65, 52, 39,.. -104 3. After how many terms does the sum of the sequence equal the following a) 6, 13, 20, 27 equal 1596 b) 18, 44, 70, 96 equal 11850 4. Find the 3 rd term of the arithmetic sequence if the 6 th term is 24 and the 15 th term is 21

Homework Questions 7 Sigma Notation 1. Rewrite the following sums using the sigma notation a) 2 + 8 + 14 + 20 +.+ 74 b) 96 + 89 + 82 + 75 + + 19 c) Multiples of 4 less than 50 d) 8 + 12 + 16 + 20 + 24 2, Calculate the following a) r = 7 r = 1 r 2 b) r = 5 r = 1 2r + 1

c) r = 10 r = 1 r 2 3 h) r = 4 r = 1 (r 6) 2 3. For what values of n does r = 1 n (n 2 + 5) first exceed 500? 4. For what value of n would r (25 6r) = 7 = 1 n

Answers Questions 1 Terms of a Sequences 1. Find the next 3 terms of the following sequences and state the rule to find the next term in each case a) 5, 9, 13, 17 21, 25, 29 b) 1, 3, 5, 7 9, 11, 13 c) 9, 13, 17, 21 25, 29, 33 d) -2, 6, 14, 22 30, 38, 46 e) 15, 22, 31, 42, 55 78, 87, 106 f) 4, 13, 26, 43, 64 89, 118, 151

g) 3, 7, 13, 21, 31 43, 57, 73 h) 5, 12, 21, 32, 45 60, 77, 96 i) 1, 2, 2, 4, 8 32, 256, 8192 j) 1, 3, 6, 10, 15 21, 28, 36

Questions 2 Using the Nth Term of A Sequences 1. Find the value of U 1, U 2, U 3 and U 20 a) U n = 3n 3, 6, 9 b) U n = 7n 2 5, 12, 19 c) U n = 2n 2 2, 8, 18 d) U n = n 2 4-3, 0, 5 2. A sequence is generate according to the formula U n =an-b. Given that U 3 =7 and U 5 =13.find the value of a and b a=3, b=2 3. Find the value of n for which U n =(3n-2) 2 has the given value of U n =100 n=4 4. A sequence is generated from the formula U n =pn 2 -q where p and q are constants. Given that U 1 =- 1 and U 3 =7, find the value of the constants p and q.

p=1 q=2 5. Find the value of n for which U n has the given value a) U n =4n-1 and U n =23 n=6 b) U n = 2n3 1 3 and U n = 5 n=2 c) U n = 5n + 6 and U n = 31 n=5

Questions 3 Recursive Formula 1. Find the next 3 terms of the following sequences given both the first term and the recursive formula. a) U 1 = 5 U n + 1 = 3U n 15, 45, 135 b) U 1 = 3 U n + 1 = 2U n -6, -12, -24 c) U 1 = 2 U n + 1 = 3U n 4-24, -76, -232 d) U 1 = 16 U n + 1 = U n 4 4, 1, 0.25 2. By writing down the first 4 terms or otherwise, find the recursive formula that defines the following sequence. a) U n =2n-1

U n+1 =U n +2 b) U n =3n-2 U n+1 =U n +3 3. Find the next 4 terms of these recursively defined sequences a) U n+1 =U n -U n-1 when U 1 =6 and U 2 =2 6, 2, 8, 10, 18, 28 b) U n+1 =3U n +2U n-1 when U 1 =1 and U 2 =-3 1, -3, -7, -27, -95, -339 c) U n+1 =5U n -11 when U 1 =3 3, 4, 9, 34, 159 4. Write down the first 3 terms of the sequence defined by U n+1 =12-U n when U 1 =10 10, 2, 10

Questions 4 General Term of an Arithmetic Sequence 1. Which of the following sequences are arithmetic? a) 7, 17, 27, 37 d=10 so yes b) 12, 5, 0, -9, -17 No c) 24, 15, 6, -3, -12 d=-9 so yes 2. a) Find the 10 th term and b) Find the formula for the nth term a) 4, 7, 10, 13 31 U n =3n+1 b) -3, -1, 1, 3 15 U n =2n-5 c) 1, -4, -9, -13

-44 U n =-5n+6 3. Find the 20 th term, if the sequence begins a) 2, 6, 10, 14, 18 78 b) 5, -3, -11, -19-147 c) 21, 27.5, 34, 40.5, 47 144.5 4. Find the number of terms in the arithmetic sequence 4, 9, 14, 19 169 34

Questions 5 Arithmetic Sequences 1. Find the number of terms in the following sequences if you are given the first few and the last term. a) 12, 25, 38,..155 12 b) 198, 192, 186, 180,.78 21 2. Find the first term of the sequence and the common difference if a) U 2 = 2 U 5 = 17 d=5 a=-3 b) U 4 = 10 U 8 = 6 d=1 a=-13 3. Find the 22 nd term and the nth term of the following sequences a) 5, 11, 17, 23.. 131 U n =6n-1

b) 25, 21, 17, 13. -59 U n =-4n+29 4. If the first term of an arithmetic sequence is 8 and the common difference is -5. what is the 22 nd term? -97 5. An arithmetic sequence has a first term of 15 and the 8 th term is 43. What are the first four terms of the sequence? 15, 19, 23, 27 6. The first two terms of an arithmetic sequence are a+2b and 7b. Find the 3 rd term. 12b-a 7. What is the common difference of the arithmetic sequence with a 6 th term of -56 and an 11 th term of 11? d=9

Questions 6 Partial sums of Arithmetic Sequences 1. Find the sum of the following series a) 17, 25, 33, 41 (25 terms) 732 b) 15, 26, 37, 42.(15 terms) 906 c) 143, 130, 117, 104.(22 terms) 858 d) 96, 90.5, 85, 79.5..(21 terms) 789 2. Find the sum of the following arithmetic sequences if you are given the first and the last term

a) 5, 19, 33, 47,..243 N=18 2232 b) 271, 263, 255, 247,..95 N=23 4209 c) 78, 65, 52, 39,.. -104 N=15-195 3. After how many terms does the sum of the sequence equal the following a) 6, 13, 20, 27 equal 1596 21 b) 18, 44, 70, 96 equal 11850 30 4. Find the 3 rd term of the arithmetic sequence if the 6 th term is 24 and the 15 th term is 21 a = 25 2 3 d = 1 3 3rd term = 25

Homework Questions 7 Sigma Notation 1. Rewrite the following sums using the sigma notation a) 2 + 8 + 14 + 20 +.+ 74 r = 13 r = 1 6r 4 b) 96 + 89 + 82 + 75 + + 19 r = 12 r = 1 7r + 103 c) Multiples of 4 less than 50 r = 12 r = 1 4r d) 8 + 12 + 16 + 20 + 24 r = 5 r = 1 4r + 4 2, Calculate the following a) r = 7 r = 1 r 2 140

b) r = 5 r = 1 2r + 1 35 c) r = 10 r = 1 r 2 3 355 h) r = 4 r = 1 (r 6) 2 54 3. For what values of n does r = 1 n (n 2 + 5) first exceed 500? n=11 4. For what value of n would r = 1 n (25 6r) = 7 n=7

Functions Mapping A function is similar to a number machine or formula, in that you put values in to the function and out come new values. The input is x, the output is f(x). Mapping is the way of showing the results from a function. It is basically saying if you have a value of x, what would be the value of the function f(x). Take a function f:x 6x + 2, (which can also be written as, f(x) = (6x + 2). If we input x = 2, we get an output of 14. This means that the function f maps 2 to 14, or 2 14. This diagram shows how each input value of x maps to only one output value of f(x). This type of mapping is called 'one-to-one mapping'. There are certain functions where many values of x give the same value of f(x). This is called 'many-to-one mapping'. An example that shows this is: f:x x 2 + 3x 2. This function maps 0 to -2 and -3 to -2. So, some values of f(x) come from more than one input value of x.

Domain and Range We need to be able to state which values of x produce values for the function f(x) and the set of these values is called the domain. Using the example function shown above f:x x 2 + 3x 2. This function produces solutions for any value of x. This means that x can be any real number - (in Further Maths there are such things as imaginary numbers!). This means we write that for, f:x x 2 + 3x 2, x R (where R is the set of real numbers). The set of output solutions produced by the function is called the range of the function. Taking the example above, we have already noted that its minimum is at x = -17/4 Therefore all the output values of the function are greater than or equal to -17/4, which is written as: Using a table At times you will need to sketch a function to see what it looks like. An easy way of doing this is: 1.

Select values of x and then calculate the corresponding values of the function. 2. Put these values in a table. 3. Use this table to sketch the graph. Using the above example, where f:x x 2 + 3x 2. Select values of x and put the corresponding values of f(x) and into an organized table: x -5-4 -3-2 -1 0 1 2 3 4 5 f(x) 8 2-2 -4-4 -2 2 8 16 26 38 Now we can plot the values of f(x) on a graph, we can see a pattern in the values of f(x): There are several important pieces of information about the function that need to be found. In particular where the graph crosses the x- and y-axes, and where the graph turns. The graph shows us that: a) The curve has a line of symmetry at the line (because values of x that are symmetrical about the line x = -3/2, give the same value for f(x)). b) The lowest value of y = -17/4 and this happens when

c) Using the quadratic formula,...we can calculate the roots of this equation (where f(x) = 0). So, And, Quadratics All quadratics have this same symmetrical shape and for a general quadratic function in the form, f(x) = ax 2 + bx + c Where a, b, and c are constants. The main features we need to sketch a quadratic are: 1. Where the graph crosses the y-axis. (At (0, c) as when x = 0, y = c). 2. Where the graph crosses the x-axis. (Factorise or use the quadratic formula to solve f(x) = 0.) 3. Where the graph turns. You can use differentiation, or completing the square (the quadratic formula), to find that:

Graphically, we see that this means: Once you know this information you can sketch any quadratic function. For example: Sketch the curve that represents f(x) -x 2 + 2x When x = 0, y = 0. Therefore it crosses the y-axis at (0,0) f(x) = 0 when -x 2 + 2x = 0, or x(2 - x) = 0. For instance, when x = 0, or when x = 2. It is a - x 2 therefore it is a symmetrical shape, with its maximum value when x = 1 (a = -1, b = 2, therefore -b/2a = 1) and y = 1. So, the graph can be sketched as:

More Complex Graphs If we don't already know what a graph will look like we need to find its main features. These are: 1. Where the graph crosses the y-axis, which is when x = 0. (i.e. at the constant). 2. Where the graph crosses the x-axis. To find the roots (where the graph crosses the x-axis), we solve the equation y = 0 3. Where the stationary points are. The stationary points occur when the gradient is 0. (i.e. differentiate.) Whether there are any discontinuities. 4. Are there any discontinuities? A discontinuity occurs when the graph is undefined for a certain value of x. This occurs when x appears in the denominator of a fraction (you can't divide by zero). 5. What happens as x approaches ±? When x becomes a large positive or a large negative number the graph will tend towards a certain value or pattern. Now put all this information onto the graph and join up the points. Example 1: Sketch the graph

If x = -3 then the denominator is zero. As we cannot divide by zero the graph is undefined, and there is a discontinuity at x = 3. As x +, y 2 (The -1 and +3 become insignificant.) As x -, y 2 as well. This means there is a horizontal asymptote (value that the graph tends towards) at x = 2. So the final graph looks like this: Example 2: The graph of the function f(x) = 2/x looks like this: The two asymptotes are the x-axis and y-axis. This curve has a special discontinuity at x = 0 where f(0) is undefined.