Geometric sequences

Elementary combinatorics: Arithmetic and geometric sequences

Geometric sequences

Geometric row

A geometric sequence with ratio \(\boldsymbol{r}\) is a sequence of numbers \(a_0,a_1,a_2, \ldots\) for which the quotient \(a_{k+1}/a_k\) of two consecutive terms of the sequence is equal to the constant \(r\). So each term in the sequence, except the first one, is the result of multiplication of its predecessor by \(r\): \(a_{k+1}=r\cdot a_k\).

Examples \[\begin{aligned}&1,2,4,8,16,\ldots\\[0.25cm] &2,6,18,54,162,\ldots\end{aligned}\]

Sum formula for a geometric sequence

If \(a_0,a_1,a_2\ldots\) is a geometric sequence with ratio \(r\), then the partial sum of the first \(n\) terms is given by \[\begin{aligned}\sum_{k=0}^{n-1}a_k&= a_0\cdot \sum_{k=0}^{n-1}r^k\\[0.25cm] &=a_0\cdot \frac{1-r^n}{1-r}\end{aligned}\]

Examples \[\begin{array}{rcrcrcrcrrc} 1&+&2&+&4&+&8&+&16&&\\ &&2&+&4&+&8&+&16&+&32\\ \hline \end{array}\] So: \(\displaystyle \sum_{k=0}^{4}2^k=\frac{1-32}{1-2}=2^5-1=31\)

If \(a_1=3, r=2, n=20\), then \(\phantom{\qquad\qquad\qquad}\) \[\begin{aligned}\sum_{k=0}^{19}(3\times 2^k) &=3\times\frac{1-2^{20}}{1-2}\\[0.25cm]&=3\times (2^{20}-1)\\[0.25cm] &=1048575\end{aligned}\]

The proof of the sum formula for the geometric sequence \(a_0\), \(a_0 r\), \(a_0 r^2\), \(\ldots\), i.e. for \(a_k=a_0 r^k\), goes as follows. Denote \(\displaystyle s_n=\sum_{k=0}^{n-1} a_0 r^k\). Then: \(r\, s_n=\sum_{k=1}^{n} a_0 r^k\) and therefore \((1-r)s_n=a_0-a_0r^n)\) or \(s_n=a_0(1-r^n)/(1-r)\).

Starting with row index 1 You can also start the indexing of a sequence of numbers with \(1\). The sum formula then becomes \[\begin{aligned}\sum_{k=1}^{n}a_k&=\sum_{k=1}^{n}a_1r^{k-1}\\[0.25cm] &=a_1\frac{1-r^{n}}{1-r}\end{aligned}\] with \(r\) the quotient between two consecutive terms from the sequence. You can see that the formula is pretty much the same as the one with indexing from \(0\). But it is a bit more difficult to describe the individual terms mathematically and the indexing does not fit the exponent of the ratio very well.