Introduction to sequences and series
A sequence is a listing of numbers. For example, 2, 4, 6, 8, or 1, 3, 5, ... are the sequences of even positive integers and odd positive integers, resp. Definition of Sequence. An infinite sequence is a function whose domain is the set of positive integers. The function values are the terms of the sequence. When the domain of the function consists of the first n positive integers only, the sequence is a finite sequence. On occasion, it is convenient to begin subscripting a sequence with 0 instead of 1 so that the terms of the sequence become
Examples of sequences Suppose that the nth term of a sequence is Write the first six terms of the sequence beginning with n = 1. Suppose another sequence is defined recursively as: Write the first six terms of the sequence. In fact, bn = n! = 1∙2 3∙4∙ ∙ ∙(n–1)∙n with 0! = 1 by definition.
Definition of Summation Notation
A convenient notation for the sum of the terms of a finite sequence is called summation notation or sigma notation. Definition of Summation Notation The sum of the first n terms of a sequence is represented by where i is called the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation. Examples.
Definition of Series Consider the infinite sequence
The sum of the first n terms of the sequence is called a finite series or the nth partial sum of the sequence and is denoted by The sum of all the terms of the infinite sequence is called an infinite series and is denoted by Example. For the series , find (a) the third partial sum and (b) the sum (a) (b)
Write the first 5 terms, begin with n = 1.
2. 3. 4. Write an expression for the nth term, an, begin with n = 1. 5. Find the sum. 6. Use sigma notation for the sum. 7. Find the sum = p/q, p and q integers.
Arithmetic sequences and partial sums
A sequence is arithmetic when the differences between consecutive terms are the same. Example. The sequence of odd numbers 1, 3, 5, 7, ... is arithmetic. What is the common difference? Example. The sequence of squares 1, 4, 9, 16, ... is not arithmetic. Why not? Example. Is the following sequence arithmetic, and if so, what is the common difference?
The nth term of arithmetic sequence
The nth term of an arithmetic sequence has the form where d is the common difference and a1 is the first term. Problem. Write a formula for the nth odd number. The first odd number is a1 = 1 and the common difference is d = 2, so the formulas is Problem. Write a formula for the nth term of the arithmetic sequence Since a1 = it follows that
An application of arithmetic sequences to simple interest
When an account earns simple interest, the balance in the account forms an arithmetic sequence. Example. Suppose you put $1000 in an account at 10% per year simple interest. What will the balance an be in the account after n years? Each year the account earns $0.10(1000) = $100 interest, so that What would the balance be if you got compound interest?
Working with arithmetic sequences
Problem. Find a formula for an for the arithmetic sequence if a1 = –4 and a5 = Solution. a5 = a1 +(5 – 1)d => 16 = –4 + 4d => d = 5. Therefore, an = –4 + (n –1)5 = –9 +5n. Given that an arithmetic sequence satisfies write the first five terms of the sequence.
Sum of a finite arithmetic sequence applied to total sales
Example. A company sells $160,000 worth of printing paper during its first year. They increase annual sales of printing paper by $20,000 each year for 3 years. What are their total sales of printing paper for their first 4 years? Let the annual sales in year i be ai . Here, a1 = 160,000 and d = 20,000 and The total sales for the 4 year period are A formula for the sum of a finite arithmetic sequence is given on the next slide.
A formula for the sum of a finite arithmetic sequence.
If is a finite arithmetic sequence with n terms, then Example from previous slide. Problem. Evaluate the sum Solution. a1 = 5 and a100 = 302.
Partial sum of an arithmetic sequence
The sum of the first n terms of an infinite sequence is called the nth partial sum. Example. Find the sum of the first 100 odd numbers. This is the 100th partial sum of the arithmetic sequence of odd numbers. For this sequence, ai = 2i – 1, and Example. Find the sum of the first 100 even numbers. This is the 100th partial sum of the arithmetic sequence of even numbers. For this sequence, ai = 2i, and
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