Tutorial on sequences and summations.
A - Introduction to Sequences
What is a sequence? A sequence may be thought of a list of ordered numbers. The sequence (or ordered list) may be finite or infinite.
Example 1:
2, 4, 6, 8, 10, ...
Example 2:
1/3, 1/5, 1/7, 1/9, 1/11, ...
We use subscripts to write the terms of a sequence as follows
a 1, a 2, a 3, ...,a n, ...
If we refer to example 1, a 1 = 2, a 2 = 4, a 3 = 6 and so on. a n is the n th term.
A sequence may also be thought of a function whose domain is the set of positive integers and may therefore be defined by a formula that gives the n th term.
Example 3: The first 4 terms of a sequence defined by a n = 2n + (-1) n are:
a 1 = 2(1) + (-1) 1 = 1
a 2 = 2(2) + (-1) 2 = 5
a 3 = 2(3) + (-1) 3 = 5
a 4 = 2(4) + (-1) 4 = 9
A sequence may also be defined recursively.
Example 4: Find the first 5 terms of a sequence defined recursively as follows:
a 1 = 1, a 2 = 3, a n = 2 a n-1 - a n-2 for n > 2
a 1 = 1 , given
a 2 = 3 , given
a 3 = 2 a 3-1 - a 3-2 = 2 a 2 - a 1 = 2(3) - 1 = 5
a 4 = 2 a 4-1 - a 4-2 = 2 a 3 - a 2 = 2(5) - 3 = 7
a 5 = 2 a 5-1 - a 5-2 = 2 a 4 - a 3 = 2(7) - 5 = 9
B-Summation Notation.
Let a 1, a 2, a 3, ..., a n be the terms of a finite sequence. A convenient notation for the sum of the terms of the sequence uses the sigma notation since it uses the Greek letter sigma, written as ∑.
Example 5: The sum of the n terms in the sequence a 1, a 2, a 3, ..., a n is written as
| n |
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∑ ai = a1 + a2 + a3 + ... + a n
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| i = 1 |
Example 6: Find the sum S of the first 7 terms in the sequence 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
S = 2+4+6+8+10+12+14 = 56
Example7: Find the sum S of the first 5 terms in the sequence whose term a n is defined by a n = (2n+1)/n.
| |
5 |
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| S= |
∑ (2n+1)/n = (2*1+1)/1 + (2*2+1)/2 + (2*3+1)/3 + (2*4+1)/4 + (2*5+1)/5 = 737 / 60
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| |
i = 1 |
Exercises:
1. What are the first 4 terms of a sequence defined by an = n2 - n?
2. Find the first 6 terms of a sequence defined recursively as follows:
a 1 = -1, a 2 = -3, a n = - a n-1 - 2 a n-2 for n > 2
3. Find the sum S of the first 5 terms in the sequence whose term a n is defined by a n = n2 + (n + 1)/2.
Answers to above exercises
1. 0, 2, 6, 12
2. -1, -2, 4, 0, -8, 8
3. 65
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