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Introduction to Sequences and Series

Posted by on October 16, 2018

 

 

Arithmetic Sequences

An arithmetic sequence is a sequence in which the difference between each consecutive term is constant. An arithmetic sequence can be defined by an explicit formula in which an = d (n - 1) + c, where d is the common difference between consecutive terms, and c = a1. An arithmetic sequence can also be defined recursively by the formulas a1 = can+1 = an + d, in which d is again the common difference between consecutive terms, and c is a constant.

 

The sum of an infinite arithmetic sequence is either ∞, if d > 0, or - ∞, if d < 0.

There are two ways to find the sum of a finite arithmetic sequence. To use the first method, you must know the value of the first term a1 and the value of the last term an. Then, the sum of the first n terms of the arithmetic sequence is Sn = n(). To use the second method, you must know the value of the first term a1 and the common difference d. Then, the sum of the first n terms of an arithmetic sequence is Sn = na1 + (dn - d ).

 

2 + 5 + 8 + 11 + 14 = 40
14 + 11 + 8 + 5 + 2 = 40

16 + 16 + 16 + 16 + 16 = 80

Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers (2 + 14 = 16). Thus 16 × 5 = 80 is twice the sum.

The sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the sum:

{\displaystyle 2+5+8+11+14}

2 + 5 + 8 + 11 + 14

This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2:

{\displaystyle {\frac {n(a_{1}+a_{n})}{2}}}

\frac{n(a_1 + a_n)}{2}

In the case above, this gives the equation:

{\displaystyle 2+5+8+11+14={\frac {5(2+14)}{2}}={\frac {5\times 16}{2}}=40.}

2 + 5 + 8 + 11 + 14 = \frac{5(2 + 14)}{2} = \frac{5 \times 16}{2} = 40.

This formula works for any real numbers 
{\displaystyle a_{1}}

a_{1} and 
{\displaystyle a_{n}}

a_{n}. For example:

{\displaystyle \left(-{\frac {3}{2}}\right)+\left(-{\frac {1}{2}}\right)+{\frac {1}{2}}={\frac {3\left(-{\frac {3}{2}}+{\frac {1}{2}}\right)}{2}}=-{\frac {3}{2}}.}

\left(-\frac{3}{2}\right) + \left(-\frac{1}{2}\right) + \frac{1}{2} = \frac{3\left(-\frac{3}{2} + \frac{1}{2}\right)}{2} = -\frac{3}{2}.

 

 

 

6A140815-C540-489F-9C4A-21E328FB9D0D E7FB71CC-7982-4693-B7C3-58533A5BD72B D0852507-DF5E-4F3C-AFCA-31BE10195309 861BBE7A-66C1-49BE-932F-E8FD7C893F1D333E17BD-15D6-4292-8A90-CC6C82954175A5740594-F4D8-4E98-A4A5-C991398E6B5CSequence – a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts at one

finite sequence – has a last term

infinite sequence – continues without stopping

 

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