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.