MATH10 SEQUENCE AND SERIES
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The document defines a sequence as a function whose domain consists of consecutive positive integers starting from 1. Each term is denoted by a(n) where n is the position in the sequence. Sequences can be finite, ending at a set number of terms, or infinite, continuing indefinitely. Examples show common sequences and their terms. The rest of the document provides examples of finding terms of sequences given their general nth term rules and prompts activities to practice finding early terms using stated rules.
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- An arithmetic sequence is one where the difference between successive terms is always the same number called the common difference.
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This document discusses arithmetic and geometric sequences and series. It provides formulas for calculating the sum of terms in an arithmetic sequence as (n/2)(a1 + an) and the sum of an arithmetic series as (n/2)(2a1 + (n-1)d) where a1 is the first term, an is the last term, n is the number of terms, and d is the common difference. It also provides a formula for calculating the sum of the first n terms of a geometric sequence as (a1(1-rn))/(1-r) where a1 is the first term, r is the common ratio, and n is the number of terms.
Task 4
The document discusses various topics related to sequences including:
- Definitions of sequences and different types of sequences such as arithmetic progressions, geometric progressions, and recurrence relations.
- Examples of sequences used in computer programming to determine if a number is even or odd through modulo operations.
- How the principle of mathematical induction can be used to prove statements about sequences, including the first and second principles of mathematical induction.
Sequences
1. The document defines sequences as sets of numbers arranged in order. Each number is called a term and is identified by its subscript position.
2. It provides examples of calculating terms of sequences using a general nth term formula or a recursive formula. Arithmetic sequences are defined as sequences where the difference between consecutive terms is constant.
3. Geometric sequences are sequences where each term is obtained by multiplying the previous term by a constant ratio. Formulas are given for calculating the nth term and summing terms of arithmetic and geometric sequences.
Sequences, Series, and the Binomial Theorem
An infinite sequence is a function whose domain is the set of natural numbers, while a finite sequence has a domain of natural numbers up to some limit. A sequence can be described by its general term, which gives a rule for calculating each term based on its position in the sequence. The sum of the terms of a sequence is called a series, which is finite if it includes a finite number of terms and infinite if it includes all terms.
6.sequences and series Further Mathematics Zimbabwe Zimsec Cambridge
This document introduces the mathematical technique of proof by induction. It uses the analogy of an inventor testing a climbing robot on ladders of varying lengths to illustrate the two conditions necessary for the robot to climb any ladder: 1) it must be able to reach the first rung, and 2) if it has reached a rung, it must be able to reach the next rung. If the robot satisfies these two conditions, it can climb infinitely high by repeatedly satisfying condition 2 after condition 1. This climbing robot analogy demonstrates how proof by induction works - it proves that a statement is true for all positive integers by showing it is true for the first integer, and then showing that if it is true for any integer n, it must
Sequence and series
SEQUENCE AND SERIES
SEQUENCE
Is a set of numbers written in a definite order such that there is a rule by which the terms are obtained. Or
Is a set of number with a simple pattern.
Example
1. A set of even numbers
• 2, 4, 6, 8, 10 ……
2. A set of odd numbers
• 1, 3, 5, 7, 9, 11….
Knowing the pattern the next number from the previous can be obtained.
Example
1. Find the next term from the sequence
• 2, 7, 12, 17, 22, 27, 32
The next term is 37.
2. Given the sequence
• 2, 4, 6, 8, 10, 12………
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MATH10 SEQUENCE AND SERIES
- 2. Sequence - a function whose domain consists of consecutive positive integers starting from 1. If a is a sequence, its terms are the values a(1) , a(2) , a(3), . . . with a(1) as the first term, a(2) as the second term, and so on.
- 3. Examples of Sequences • 1, 4, 7,10, 13 • 0, 1, 2, 0, 1, 2, 0, 1, 2 • 0, 5, 10, 15, 20, 25, . . .
- 4. Infinite Sequence - sequence whose domain is the set of all positive integers. Finite Sequence - sequence whose domain is the first n positive integers.
- 5. General Terms of Sequences EXAMPLE NO. 1 The nth term of a sequence with 20 terms is denoted by an = 3n + 1. Find its first four terms and its twentieth term.
- 8. General Terms of Sequences EXAMPLE NO. 2 The nth term of an infinite sequence is denoted by an = (-1)n • 5 𝑛 Find a1 , a2, a5 and a10
- 11. EXAMPLE NO. 3 The terms of a sequence are described by the following rule: a1 = 5 an = an – 1 -2 if n > 1 Find a2, a3 , a4, and a5
- 13. ACTIVITY NO. 1
- 14. 𝑎𝑛 = 3𝑛 𝑎𝑛 = 2 − 𝑛 𝑎𝑛 = 4𝑛 − 1 𝑎𝑛 = 3𝑛 − 1 𝑎𝑛 = 1 𝑛 + 1 Given the nth term an , find the first four terms of each sequence. ACTIVITY NO. 1
- 15. 𝑎𝑛 = 3𝑛 − 1 Given the nth term an , find the first four terms of each sequence. ACTIVITY NO. 1
- 16. 𝑎1 = 2 and 𝑎𝑛 = 𝑎𝑛−1 + 3 𝑓𝑜𝑟 𝑛 > 1 Find the first three terms of each sequence using the given rule. 𝑎1 = 5 and 𝑎𝑛 = 2𝑎𝑛−1 − 1 𝑓𝑜𝑟 𝑛 > 1
- 17. 𝑎1 = 2 and 𝑎𝑛 = 𝑎𝑛−1 + 3 𝑓𝑜𝑟 𝑛 > 1
Editor's Notes
- Finite sequence have its last term while infinite sequence does not have last term