# 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}$