Chapter 6 Sequences and Series. Arithmetic and geometric sequences and series. The sequence defined by u1 = a and un = un−1 + d for n ≥ 2 begins. PDF Drive is your search engine for PDF files. As of today we have MOOCULUS massive open online calculus sequences and series. THIS DOCUMENT WAS. We begin by discussing the concept of a sequence. Intuitively, a sequence is an ordered list of objects or events. For instance, the sequence of events at a crime.
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Sequences and Series and then lim i→∞1 −. 1. 2i= 1 − 0=1. There is one place that you have long accepted this notion of infinite sum without . mcTY-apgp This unit introduces sequences and series, and gives some simple examples of each. It also explores particular types of sequence known. Mathematical Induction. 8. Sequences. Series. Power Series. Taylor Series. Solutions to the exercises in this booklet are available at the Web-site .
About this book Introduction This book aims to dispel the mystery and fear experienced by students surrounding sequences, series, convergence, and their applications. The author, an accomplished female mathematician, achieves this by taking a problem solving approach, starting with fascinating problems and solving them step by step with clear explanations and illuminating diagrams. The reader will find the problems interesting, unusual, and fun, yet solved with the rigor expected in a competition. Some problems are taken directly from mathematics competitions, with the name and year of the exam provided for reference. Proof techniques are emphasized, with a variety of methods presented. The text aims to expand the mind of the reader by often presenting multiple ways to attack the same problem, as well as drawing connections with different fields of mathematics. Intuitive and visual arguments are presented alongside technical proofs to provide a well-rounded methodology.
Find the sum of the first n integers. Find the sum of the fist n odd numbers, simplifying your answer as far as possible.
Find the sum of the fist ten multiples of 7 which are greater than Find x. Find x and y. After each 2 bounce the time until it returns to each is multiplied by a factor of. And so on. Find the n th term for each of the sequences in Question 1. Find Expressions for the sum to n terms of each of the sequences in Question1. Find, where possible , the sum to infinity of the sequence in Question 1.
The fourth term of a GP is 4 and the sixth is Find two possible values for the common ratio, and hence find two possible value for the first term. A GP has first term 5 and common ratio 4. How many terms of the sequence are less than 9, ? A GP has first term 2 and common ratio 1. How many term of the sequence most be taken before their sum is greater than ? A GP has first term 6 and common ratio 0. Which is the first term of the sequence less than 1?
Solution Let the first term be a , the common difference be d and the common ratio be r.
Find the common ratio. Solution Let the first term be a and the common ratio r.
How much will be have at the beginning of the 13 th year ,i. Solution The which was originally invested to. The total sum of money will be the sum of 13 terms of a GP , with first term and common ratio 1.
The third term of an AP is x , and the seventh term is y. Express the common difference and the first term as functions of x and y. Find the possible values of x. Find the sum of all the numbers less than whose last digit is 5.
How much will be have in the bank just before his tenth investment? A job is advertised at a starting salary of 12, with annual increments of How much will the salary be during the 10 th years? How much will be the total sum earned after 10 years? What will salary be during the 10 th years? What will the total sum earned after 10 years? A child builds a triangular wall of bricks , with 1 brick on top now , 2 brick on the second , 3 on the third and so on.
If the wall is 8 bricks high, how many bricks does it contain? If the child has only 70 bricks ,what is the tallest wall she can build? A large sheet of paper is 0. How thick is it after it has been folded ten time? Assuming that this is possible. How many times must it be folded before it is 1 metre thick 9.
A king was so pleased with the game of chess that he granted its inventor any prize he desired. The inventor asked for one grain of wheat on the first square, two grain on the second square, four grains on the third square and so on for all 64 square. How many grains is he asking for in total? Let x be the recurring decimal 0. By the method of Question 11 express the following as fraction in their simplest form: a 0. Achilles can run at 10 feet per second, and the Tortoise at 1 foot per second.
Write down the next four terms of each sequence: a 9, 13, 17,. Give a recursive formula for the nth term Tn of each sequence in terms of the n 1 th term Tn 1 : a 16, 21, 26,.
The rigorous denition of a sequence is: A sequence is a function whose domain is the set of positive integers. Graph the sequences in question 2, with n on the horizontal axis and Tn on the vertical axis. If there is a simple curve joining the points, draw it and give its equation. Find T3 and T4. Explain why every third term of each sequence is even and the rest are odd. How are these two new sequences related to the Fibonacci sequence, and why?
This is a sequence like 3, 7, 11, 15, 19, 23, 27, 31, 35, 39,. Because the dierence is constant, all the terms can be generated from the rst term 3 by repeated addition of this common dierence 4.
Arithmetic sequences are called APs for short, standing for arithmetic progression, an old name for the same thing. Denition of an Arithmetic Sequence: An arithmetic sequence is a sequence in which the dierence between successive terms is constant. Then nd a formula for the nth term, and nd the rst negative term.