Series of successions of numbers that are formed by a common law, meaning each number is formed by the same arithmetic operation on the former.
Important aspects of sequences or progressions are:
- The law of its formation
- The sum of a given number of its terms
Arithmetic Progressions
Are sequences formed by addition or subtraction of each number with a fixed constant, called the common difference of the series.
Examples
1) , common difference:
2) , common difference:
Therefore if are in arithmetic sequence:
That means, subtracting a number by its predecessor yields the common difference.
Describing any Term in the Sequence
if and :
This implies the formula for the th term of the progression:
Last Term of a Sequence
Sum of any Number of Terms
Arithmetic Mean
Given three numbers in any arithmetic progression, the middle term is always their arithmetic mean.
Harmonic Progressions
Are series of reciprocals:
If is a series in arithmetic progression, then
is a series in harmonic progression.
Sums of harmonic sequences can be found using their arithmetic counter parts.
Harmonic Mean
The mean of a harmonic series is found using the corresponding arithmetic sequence, and finding the reciprocal of the arithmetic mean.
Geometric Progressions
Are series where each number is a multiple of its preceding number and a common ratio.
Rule
If is a series in geometric progression:
So each term over the previous one is the same ratio.
Note that the logarithms of a geometric sequence form a series in arithmetic progression:
Formula for the th term of a geometric series:
Geometric Mean
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Sum of any Number of Terms
If :
Else ():
Increasing Sequences
With a common ratio of or greater the sum of a geometric progression increases with the number of terms .
Therefore if increases without limit, so does their sum, meaning both increase towards infinity.
Decreasing Sequences
If as grows towards infinity the value of the terms becomes increasingly low - they decrease indefinitely. Therefore there is a limit to their sum.
Consider that is in effect a geometric sequence:
So no matter how large may become, the sum never exceeds .
The Sum to a Limit
The sum approaches the limit as increases towards infinity.
Therefore given a ratio , the following formula calculates the limiting sum of a sequence: