Arithmetic series

Arithmetic series are special mathematical series . An arithmetic series is the sequence whose terms are the sum of the first terms (the partial sums ) of an arithmetic sequence . Arithmetic series are generally divergent. The partial sums, which are also called finite arithmetic series , are of particular interest . ${\displaystyle n}$

In an arithmetic sequence , the -th term can be used as ${\displaystyle (a_{i})_{i\in \mathbb {N} }}$ ${\displaystyle \ i}$ ${\displaystyle \ a_{i}}$

${\displaystyle a_{i}=a_{1}+(i-1)d}$

write, where is the (constant) difference between two consecutive terms. ${\displaystyle d=a_{i+1}-a_{i}}$

The -th partial sum of an arithmetic series results in ${\displaystyle n}$ ${\displaystyle s_{n}}$

${\displaystyle {s_{n}=a_{1}+a_{2}+\dotsb +a_{n}=\sum _{i=1}^{n}a_{i}}}$.

General molecular formula

There is a simple formula for calculating the partial sums (or the finite arithmetic series):

${\displaystyle {s_{n}=\sum _{i=1}^{n}(a_{1}+(i-1)d)=na_{1}+{\frac {n(n-1)}{2}}\,d={\frac {n}{2}}(2a_{1}+(n-1)d)=n\cdot {\frac {a_{1}+a_{n}}{2}}}}$.

In the last form, the formula is particularly easy to remember: The sum of a finite arithmetic sequence is the number of terms multiplied by the arithmetic mean of the first and the last term.

The proof of this equation is often used as a first example of the application of the complete induction method.

Special sums

The Gaussian empirical formula applies to the sum of the first natural numbers ${\displaystyle n}$

${\displaystyle \sum _{k=1}^{n}k=1+2+3+\dotsb +n={\frac {n(n+1)}{2}}}$

and for the sum of the first odd natural numbers ${\displaystyle n}$

${\displaystyle \sum _{k=1}^{n}(2k-1)=1+3+5+7+\dotsb +(2n-1)=n\cdot {\frac {1+2n-1}{2}}=n^{2}}$

with , . ${\displaystyle a_{1}=1}$ ${\displaystyle d=2}$

See also

• Geometric series
• Binomial series
• Sequence of differences

Web links

• Eric W. Weisstein : Arithmetic Series . In: MathWorld (English).