tailieunhanh - Ebook Additional mathematics: Pure and applied (6th edition): Part 2

(BQ) This sixth edition of Additional maths:Pure and applied, has been completely revised and updated. it covers the Cambridge additional mathematics syllabus in the pure mathematics and particle mechanics sections. The topics have been taken largely in the order of the syllabus for convenience, but this could be altered if desired. | Arithmetic and Geometric Progressions ARITHMETIC PROGRESSIONS Here are 3 sequences or numbers which follow a simple patiem. Can you say what the next two numbers should be i 1 10 . it -15 -11 -7 -3 . iii 29 272 26 242 . You will have found that in i the numbers increase by 3 so the next two numbers are 13 and 16 in ii the numbers increase by 4 so the next two are 1 and 5. and in iii the numbers decrease by 12 so the next two arc 23 and 212. These arc examples of an arithmetic progression which we shall abbreviate as AP. An AP is a sequence of numbers which increase by a constant amount or - . This amount is called the common difference d and the starting number is called the first term a . Hence the second term or T for short is a d. the third term T is a d d a 2d and so on. 4th term a n i d So the formula for the nth term 7 of an AP is Note that the difference between consecutive terms is constant 306 Example 1 a What is the I5th term of the AP -5 . lb Which term is 28 In this AP. a -5 and d 3. a r a I4d -5 42 37 b a n - so 28 -5 n - 1 X 3 3n - 8 Hence n 12 . 28 is the 12th term. Example 2 Find a formula in terms of n for the nth term of the AP 15 9 3. and hence find the 30th term. T a n - l 15 n - 1 X -6 21 - 6n Then TK 21 - 6 X 30 -159. Example 3 The nth term of an AP is given byT 2n Í difference. a r 2xl 9 ll b d T3- T 2x2 9-11 2 . Find la the first term b the common Example 4 If the 5th term of an AP is and the 10th term is 16 find the first term and the common difference. Ts a 4d -4 and Tla a 9d 16. Solving these equations d 4 and a -20. 307 Example 5 The first term of an AP is -4 and the 15th term is double the 5 th term. Find the 12th We must first find d. T15 -4 14d and Ts -4 4d. T J 2T so -4 14d 2 -4 4d -8 8d. Hence 6d -4 and d - 3. Then the 12th term a 1 Id - 4 11 X -I 11 J. Example 6 The sum of three consecutive terms of an AP is 18 and their product is 120. Find these We could take d and k 2d as the three consecutive terms but it is .