Arithmetico-geometric sequence

Not to be confused with Arithmetic–geometric mean.

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Specialized

In mathematics, an arithmetico-geometric sequence is the result of the multiplication of a geometric progression with the corresponding terms of an arithmetic progression. It should be noted that the corresponding French term refers to a different concept (sequences of the form $u_{n+1}=au_{n}+b$ ) which is a special case of linear difference equations.

Sequence, nth term

The sequence has the nth term^[1] defined for n ≥ 1 as:

[a+(n-1)d]br^{n-1}

are terms from the arithmetic progression with difference d and initial value a and geometric progression with initial value "b" and common ratio "r"

Series, sum to n terms

An arithmetico-geometric series has the form

\sum _{{k=1}}^{n}\left[a+(k-1)d\right]r^{{k-1}}=a+[a+d]r+[a+2d]r^{2}+\cdots +[a+(n-1)d]r^{{n-1}}

and the sum to n terms is equal to:

S_{n}=\sum _{{k=1}}^{n}\left[a+(k-1)d\right]r^{{k-1}}={\frac {a-[a+(n-1)d]r^{n}}{1-r}}+{\frac {dr(1-r^{{n-1}})}{(1-r)^{2}}}.

Derivation

Starting from the series,^[1]

S_{n}=a+[a+d]r+[a+2d]r^{2}+\cdots +[a+(n-1)d]r^{{n-1}}

multiply S_n by r,

rS_{n}=ar+[a+d]r^{2}+[a+2d]r^{3}+\cdots +[a+(n-1)d]r^{n}

subtract rS_n from S_n,

{\begin{aligned}(1-r)S_{n}&=&\left[a+(a+d)r+(a+2d)r^{2}+\cdots +[a+(n-1)d]r^{{n-1}}\right]\\&&-\left[ar+(a+d)r^{2}+(a+2d)r^{3}+\cdots +[a+(n-1)d]r^{n}\right]\\&=&a+d\left(r+r^{2}+\cdots +r^{{n-1}}\right)-\left[a+(n-1)d\right]r^{n}\\&=&a+{\frac {dr(1-r^{{n-1}})}{1-r}}-[a+(n-1)d]r^{n}\end{aligned}}

using the expression for the sum of a geometric series in the middle series of terms. Finally dividing through by (1 − r) gives the result.

Sum to infinite terms

If −1 < r < 1, then the sum of the infinite number of terms of the progression is^[1]

\lim _{{n\to \infty }}S_{{n}}={\frac {a}{1-r}}+{\frac {rd}{(1-r)^{2}}}

If r is outside of the above range, the series either

diverges (when r > 1, or when r = 1 where the series is arithmetic and a and d are not both zero; if both a and d are zero in the later case, all terms of the series are zero and the series is constant)
or alternates (when r ≤ −1).

References

1 2 3 K.F. Riley; M.P. Hobson; S.J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. p. 118. ISBN 978-0-521-86153-3.