Mathematics

Understanding the Differences between the Mathematical Subjects Sequences and Series



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Sequences can be finite or infinite. Each term of a sequence is written using subscripts (a1, a2, a3, a4,...,an), whose an is the nth term. A sequence is finite if it ends at a defined an. Formulas are used to define sequences. For example, the infinite sequence of odd positive integers 1, 3, 5, 7, 9, 11, ... is defined using an=2N+1 for N=1, 2, 3, 4, 5, 6, 7, ... . A recurrence relation uses some of the preceding terms to define a sequence.


An example of a finite sequence is an=2N for N=1, 2, 3, 4. Its members are 2, 4, 6, and 8. An arithmetic sequence, also called arithmetic progression, has a common difference between terms.

This common difference is added to the first term a. For example, if the first term of a sequence is a=5, and the common difference is d=10, then the terms start at 5 with 10 added for each successive term: 5, 15, 25, 35, 45, 55, ... . The formula for the nth term is a+(N-1)d, where a is the first term, N is the number of the nth term (first term is N=1), and d is the common difference.

The tenth term of the a=5, d=10 sequence is found using a=5, d=10, N=10 in the formula:

5+(10-1)10=95. It checks: 5, 15, 25, 35, 45, 55, 65, 75, 85, 95.


A geometric sequence (geometric progression) has a geometric ratio between terms. The geometric ratio is multiplied to the preceding term instead of added as in arithmetic sequences.

An example is if the first term of a geometric sequence is a=7 and the geometric ratio is R=3, then the sequence is 7, 21, 63, 189, ... . The formula for the nth term of a geometric sequence is aR^(N-1), where a is the first term of the sequence, R is the geometric ratio, and N is the number

of the nth term (first term is N=1).  The fourth term of the a=7, R=3 sequence is found using a=7, R=3, and N=4 in the formula aR^(N-1):  (7)(3)^(4-1)=(7)(27)=189.  It checks:  7, 21, 63, 189. 


If the terms of a sequence are added together, it is a series.  The Greek letter sigma is used to

denote “sum of”.  The sum of the terms of the arithmetic sequence with a=5, d=10, and N=10

is 5+15+25+35+45+55+65+75+85+95=500.  The sum of the terms of the geometric sequence with a=7, R=3, and N=4 is 7+21+63+189=280. 


The formula for the sum of a finite arithmetic series is N/2[2a+(N-1)d], where a is the first term, d is the common difference, and N is the number of terms.  Using this formula, the sum of the first ten terms (N=10) of the a=5, d=10, and N=10 arithmetic sequence is

10/2[2(5)+(10-1)10]=50+450=500. 


The formula for the sum of a finite geometric series is (a-aR^N)/(1-R), where a is the first term,

R is the common ratio (R not equal to 1 because of division by zero), and N is the number of terms.  Using this formula, the sum of the first four terms (N=4) of the geometric sequence

With a=7 and R=3 is [7-(7)(3)^4]/[1-3]=-560/-2=280.  The sum of infinite sequences (infinite series) is harder.  Some infinite series do not have a defined sum.   



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