Arithmetic sequences

Elementary combinatorics: Arithmetic and geometric sequences

Arithmetic sequences

Arithmetic sequence

An arithmetic sequence is a sequence of numbers $a_1,a_2,a_3, \ldots$ for which the difference $a_{k+1}-a_k$ between two successive terms of the sequence is constant.

Examples

$\begin{aligned}&1,2,3,4,5,6,\ldots\\[0.25cm] &2,5,8,11,14,\ldots\end{aligned}$

Summation formula fot an arithmetic sequence

If $a_1,a_2,a_3, \ldots$ is an arithmetic sequence, then the partial sum of the first $n$ terms is given by

$\sum_{k=1}^{n}a_k=\tfrac{1}{2}n(a_1+a_n)$ If the difference between two successive terms of the sequence is equal to $v$ , then the sum formula can also be written as

$\sum_{k=1}^{n}a_k=n\,a_1+\tfrac{1}{2}(n-1)\,n\,v$

Examples

$\begin{array}{rcrcrcrcr} 1&+&2&+&\ldots&+&99&+&100\\ 100&+&99&+&\ldots&+&2&+&1\\ \hline 101&+&101&+&\ldots&+&101&+&101\end{array}$ So: $\displaystyle \sum_{k=1}^{100}k=\frac{1}{2}\times 100\times 101=5050$

If $a_1=3, v=2, n=20$ , then $\phantom{\qquad\qquad\qquad}$

$\sum_{k=1}^{20}(2k+1)=20\times 3+\frac{1}{2}\times 19\times 20\times 2=440$

Starting with the sequence index 0 You can also start the indexing of a sequence with $0$ . The summation formula becomes in this case

$\sum_{k=0}^{n-1}a_k=\tfrac{1}{2}n(a_0+a_{n-1})$ and

$\sum_{k=0}^{n-1}a_k=n\,a_{0}+\tfrac{1}{2}(n-1)\,n\,v$ where $v$ is the difference between two successive terms of the sequence. The formulas are almost as those corresponding to indexing from $1$ . But for geometric sequences, this new indexing will turn out to be more convenient.