Partial sums The sum of the first terms of an arithmetico-geometric series has the form : \begin{align} S_n & = \sum_{k = 1}^n A_k G_k \\[5pt] & = \sum_{k = 1}^n \bigl(a + (k - 1) d\bigr) br^{k - 1} \\[5pt] & = b \sum_{k = 0}^{n-1} \left(a + k d\right) r^{k} \\[5pt] & = ab + (a + d) br + (a + 2 d) br^2 + \cdots + \bigl(a + (n - 1) d\bigr) br^{n - 1} \end{align} where A_i and G_i are the th elements of the arithmetic and the geometric sequence, respectively. This partial sum has the
closed-form expression : \begin{align} S_n & = \frac{ab - (a+nd)\,br^n}{1 - r}+\frac{dbr\,(1 - r^n)}{(1-r)^2}\\ & = \frac{A_1G_1 - A_{n+1}G_{n+1}}{1 - r}+\frac{dr}{(1-r)^2}\,(G_1 - G_{n+1}). \end{align}
Derivation Multiplying :S_n = ab + (a + d) br + (a + 2 d) br^2 + \cdots + \bigl(a + (n - 1) d\bigr) br^{n - 1} by gives :r S_n = a br + (a + d) br^2 + (a + 2 d) br^3 + \cdots + \bigl(a + (n - 1) d\bigr) br^n. Subtracting from , dividing both sides by b, and using the technique of
telescoping series (second equality) and the formula for the sum of a finite
geometric series (fifth equality) gives :\begin{align} \frac{(1 - r) S_n}{b} &= \left(a + (a + d) r + (a + 2 d) r^2 + \cdots + \bigl(a + (n - 1) d\bigr) r^{n - 1}\right) - \Bigl(a r + (a + d) r^2 + (a + 2 d) r^3 + \cdots + \bigl(a + (n - 1) d\bigr) r^n\Bigr) \\[5pt] &= a + d \left(r + r^2 + \cdots + r^{n-1}\right) - \bigl(a + (n - 1) d\bigr) r^n \\[5pt] &= a + d \left(r + r^2 + \cdots + r^{n-1}+r^n\right) - \left(a + n d\right) r^n \\[5pt] &= a + d r \left(1 + r + r^2 + \cdots + r^{n-1}\right) - \left(a + nd\right) r^n \\[5pt] &= a + \frac{d r (1 - r^n)}{1 - r} - (a + nd) r^n, \\[8pt] S_n &= \frac{b}{1-r} \left( a - (a + nd) r^n + \frac{d r (1 - r^n)}{1 - r} \right) \\[5pt] &= \frac{ab - (a + nd) b r^n}{1-r} + \frac{d r (b - br^n)}{(1 - r)^2} \\[5pt] &= \frac{A_1 G_1 - A_{n+1} G_{n+1}}{1-r} + \frac{d r (G_1 - G_{n+1})}{(1 - r)^2} \end{align} as claimed.
Infinite series If , then the sum of the arithmetico-geometric
series, that is to say, the
limit of the partial sums of the elements of the sequence, is given by : \begin{align} S &= \sum_{k = 1}^\infty t_k = \lim_{n \to \infty}S_{n} \\[5pt] &= \frac{ab}{1-r}+\frac{dbr}{(1-r)^2}\\[5pt] &= \frac{A_1G_1}{1 - r}+\frac{d r G_1}{(1-r)^2}. \end{align} If is outside of the above range, is not zero, and and are not both zero, the limit does not exist and the series is
divergent.
Example The sum :S=\frac{\color{blue}{0}}{\color{green}{1}}+\frac{\color{blue}{1}}{\color{green}{2}}+\frac{\color{blue}{2}}{\color{green}{4}}+\frac{\color{blue}{3}}{\color{green}{8}}+\frac{\color{blue}{4}}{\color{green}{16}}+\frac{\color{blue}{5}}{\color{green}{32}}+\cdots , is the sum of an arithmetico-geometric series defined by d=b=1, a=0, and r=\tfrac 12, and it converges to S=2. This sequence corresponds to the expected number of
coin tosses required to obtain "tails". The probability T_k of obtaining tails for the first time at the
kth toss is as follows: :T_1=\frac 1{2}, \ T_2=\frac 1{4},\dots, T_k = \frac 1{2^k}. Therefore, the expected number of tosses to reach the first "tails" is given by :\sum_{k=1}^{\infty} k T_k = \sum_{k=1}^{\infty} \frac {\color{blue}k}{\color{green}2^k} = 2. Similarly, the sum : S=\frac{\color{blue}{0} \cdot \color{green}\frac16}{\color{green}\frac56}+\frac{\color{blue}{1} \cdot \color{green}\frac16}{\color{green}{1}}+\frac{\color{blue}{2} \cdot \color{green}\frac16}{\color{green}{\frac65}}+\frac{\color{blue}{3} \cdot \color{green}\frac16}{\color{green}{\left(\frac65\right)^2}}+\frac{\color{blue}{4} \cdot \color{green}\frac16}{\color{green}{\left(\frac65\right)^3}}+\frac{\color{blue}{5} \cdot \color{green}\frac16}{\color{green}{\left(\frac65\right)^4}}+\cdots is the sum of an arithmetico-geometric series defined by d = 1 , a = 0 , b = \tfrac{1/6}{5/6} = \tfrac15 , and r = \tfrac56 , and it converges to 6. This sequence corresponds to the expected number of
six-sided dice rolls required to obtain a specific value on a die roll, for instance "5". In general, these series with d = 1 , a = 0 , b = \tfrac{p}{1 - p} , and r = 1-p give the expectations of "the number of trials until first success" in
Bernoulli processes with "success probability" p . The probabilities of each outcome follow a
geometric distribution and provide the geometric sequence factors in the terms of the series, while the number of trials per outcome provides the arithmetic sequence factors in the terms. ==References==