Arithmetico-geometric sequence

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an arithmetico-geometric sequence is the result of element-by-element multiplication of the elements of a <a href="/facts/Geometric_progression/SVrkRHDX">geometric progression</a> with the corresponding elements of an <a href="/facts/Arithmetic_progression/c4ew9Xoq">arithmetic progression</a>. The nth element of an arithmetico-geometric sequence is the product of the nth element of an arithmetic sequence and the nth element of a geometric sequence. An arithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various applications, such as the computation of <a href="/facts/Expected_values/1XV0JKL8">expected values</a> in <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, especially in <a href="/facts/Bernoulli_process/AF9E3jpu">Bernoulli processes</a>.
For instance, the sequence

0
              
            
            
              
                1
              
            
          
        
        ,
         
        
          
            
              
                1
              
            
            
              
                2
              
            
          
        
        ,
         
        
          
            
              
                2
              
            
            
              
                4
              
            
          
        
        ,
         
        
          
            
              
                3
              
            
            
              
                8
              
            
          
        
        ,
         
        
          
            
              
                4
              
            
            
              
                16
              
            
          
        
        ,
         
        
          
            
              
                5
              
            
            
              
                32
              
            
          
        
        ,
        ⋯
      
    
    {\displaystyle {\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 an arithmetico-geometric sequence. The arithmetic component appears in the numerator (in blue), and the geometric one in the denominator (in green). The <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a> summation of the infinite elements of this sequence has been called Gabriel's staircase and it has a value of 2. In general,

∑
 
 k
 =
 1
 
 
 ∞
 
 
 
 
 k
 
 
 
 
 
 r
 
 k
 
 
 
 
 =
 
 
 r
 
 (
 1
 −
 r
 
 )
 
 2
 
 
 
 
 
 
 
 for 
 
 −
 1
 <
 r
 <
 1.
 
 
 {\displaystyle \sum _{k=1}^{\infty }{\color {blue}k}{\color {green}r^{k}}={\frac {r}{(1-r)^{2}}}\quad {\text{for }}-1<r<1.}

The label of arithmetico-geometric sequence may also be given to different objects combining characteristics of both arithmetic and geometric sequences. For instance, the French notion of arithmetico-geometric sequence refers to sequences that satisfy <a href="/facts/Recurrence_relation/JolXC71M">recurrence relations</a> of the form 
 
 
 
 
 u
 
 n
 +
 1
 
 
 =
 r
 
 u
 
 n
 
 
 +
 d
 
 
 {\displaystyle u_{n+1}=ru_{n}+d}
 
, which combine the defining recurrence relations 
 
 
 
 
 u
 
 n
 +
 1
 
 
 =
 
 u
 
 n
 
 
 +
 d
 
 
 {\displaystyle u_{n+1}=u_{n}+d}
 
 for arithmetic sequences and 
 
 
 
 
 u
 
 n
 +
 1
 
 
 =
 r
 
 u
 
 n
 
 
 
 
 {\displaystyle u_{n+1}=ru_{n}}
 
 for geometric sequences. These sequences are therefore solutions to a special class of <a href="/facts/Linear_difference_equation/gs9PR0D7">linear difference equation</a>: inhomogeneous first order <a href="/facts/Linear_recurrence_with_constant_coefficients/gs9PR0D7">linear recurrences with constant coefficients</a>.

Arithmetico-geometric sequence open-in-new

Arithmetico-geometric sequence