7 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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Unformatted text preview: ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 7 PROGRESSIONS 7.1. Arithmetic Progressions Example 7.1.1. Consider the finite sequence of numbers 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 , 17 , 19 , 21 , 23 , 25 , 27 , 29 , 31 . This sequence has the property that the difference between successive terms is constant and equal to 2. It follows that the k-th term is obtained from the first term by adding ( k 1) 2, and is therefore equal to 1 + 2( k 1). On the other hand, if we want to add all the numbers together, then we observe that 1 + 31 = 3 + 29 = 5 + 27 = 7 + 25 = 9 + 23 = 11 + 21 = 13 + 19 = 15 + 17 , so that the numbers can be paired off in such a way that the sum of the pair is always the same and equal to 32. Note now that there are 16 numbers which form 8 pairs. It follows that 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31 = 8 32 = 256 . Example 7.1.2. Consider the finite sequence of numbers 2 , 5 , 8 , 11 , 14 , 17 , 20 , 23 , 26 , 29 , 32 . This sequence has the property that the difference between successive terms is constant and equal to 3....
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7 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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