# How are arithmetic sequences used

## Sequences and ranks¶

### Rows and their properties¶

The sum of the terms of a sequence (or part of the terms of the sequence) is called a series. Mathematical is the sum the members of a sequence by the sums symbol expressed:

Here, the lower limit is below the sum symbol and the upper limit of the index above indicated, whereby the sum limits are in each case whole numbers. In the above case, all of the sequence members are thus from to summed up.

Is the lower summation limit equal to the above, it means that the sum of a single number consists:

If the lower summation limit is greater than the upper summation limit, the result of the sum is defined as zero. Other important calculation rules for the summation sign are:

The upper two of these calculation rules correspond to a rearrangement of the summands, the last one to excluding the factor from every summand. This rule also applies when one The following terms with constant value summed up:

Digital counting machines also work according to the above equation, adding up a series of (mostly electrical) “one” signals and the corresponding value Show.

Two other computing tricks are often used to good effect when dealing with rows:

The value of a series remains unchanged by an index shift. This means a process of the following type:

In the general case, the index of the summation limits is around raised, the index of the sequential members must be on be reduced. [4] The following applies:

A reduction in the summation limit by causes an increase in the index of the following elements in a corresponding manner :

(10)¶

### Arithmetic series¶

If you add up all the terms of an arithmetic sequence, i.e. a sequence of numbers that are always the same value among each other differ, an arithmetic series results. For the value of the most well-known arithmetic series, in which all natural numbers of to are added, Carl Friedrich Gauss found the following formula at a young age, which is sometimes also called "Little Gauss": [5] [6]

(11)¶

In the general case, the value of an arithmetic series can be calculated as follows: [7]

(12)¶

### Geometric series¶

If you add up all the terms of a geometric sequence, i.e. a sequence of numbers that are always related to each other by the same factor differ, a geometric series results. The value a finite geometric series can be calculated as follows: [8]

(13)¶

Whether an infinite geometric series converges depends on the value of from. Is so the series diverges; is on the other hand , then the series converges, and we have:

For example, compound interest can be calculated using geometric series.

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