en>John of Reading: Typo/general fixing, replaced: of had → of them had using AWB

2014-01-03T11:26:33Z

Typo/general fixing, replaced: of had → of them had using AWB

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{{Multiple issues |{{expert-subject|date=December 2012|reason=Mathematics}} {{lead too short|date=December 2012}} {{Refimprove|date=November 2012}}}}
{{Calculus |Series}}

In [[mathematics]], an '''arithmetico-geometric sequence''' is the result of the multiplication of a [[geometric progression]] with the corresponding terms of an [[arithmetic progression]].

==Sequence, nth term==
The sequence has the ''n''th term<ref>{{cite book |author= K.F. Riley, M.P. Hobson, S.J. Bence |title= Mathematical methods for physics and engineering|edition= 3rd|year= 2010|page=118|publisher= Cambridge University Press|isbn=978-0-521-86153-3}}</ref> defined for ''n'' ≥ 0 as:

:<math>[a+(n-1)d] r^{n-1} </math>

where ''r'' is the [[Geometric series#Common ratio|common ratio]], and the coefficients of ''r''''n'' − 1

:<math>[a+(n-1)d]</math>

are terms from the [[arithmetic progression]] with difference ''d'' and initial value ''a''.

== Series, sum to ''n'' terms ==
An arithmetico-geometric series has the form

:<math>\sum_{k = 1}^n \left[a + (k - 1) d\right] r^{k - 1} = a + [a + d] r + [a + 2 d] r^2 + \cdots + [a + (n - 1) d] r^{n - 1}</math>

and the sum to ''n'' terms is equal to:

:<math>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}.</math>

=== Derivation===
Starting from the series,<ref>{{cite book |author= K.F. Riley, M.P. Hobson, S.J. Bence |title= Mathematical methods for physics and engineering|edition= 3rd|year= 2010|page=118|publisher= Cambridge University Press|isbn=978-0-521-86153-3}}</ref>

:<math>S_n = a + [a + d] r + [a + 2 d] r^2 + \cdots + [a + (n - 1) d] r^{n - 1}</math>

multiply ''Sn'' by ''r'',

:<math>r S_n = a r + [a + d] r^2 + [a + 2 d] r^3 + \cdots + [a + (n - 1) d] r^n</math>

subtract ''rSn'' from ''Sn'',

:<math>\begin{align}
(1 - r) S_n &=& \left[a + (a + d) r + (a + 2 d) r^2 + \cdots + [a + (n - 1) d] r^{n - 1}\right] \\
& & - \left[a r + (a + d) r^2 + (a + 2 d) 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{d r (1 - r^{n - 1})}{1 - r} - [a + (n - 1) d] r^n \end{align}
</math>

using the expression for the [[Geometric series#Formula|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<ref>{{cite book |author= K.F. Riley, M.P. Hobson, S.J. Bence |title= Mathematical methods for physics and engineering|edition= 3rd|year= 2010|page=118|publisher= Cambridge University Press|isbn=978-0-521-86153-3}}</ref>

:<math>\lim_{n \to \infty}S_{n} = \frac{a}{1-r}+\frac{rd}{(1-r)^2}</math>

If ''r'' is outside of the above range, the series either
* [[Divergent series|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 [[Alternating series|alternates]] (when ''r'' ≤ −1).

==See also==
*[[Partial sum]]

==References==
{{reflist}}

*{{cite book|title=The Pearson Guide to Mathematics for the IIT-JEE, 2/e (New Edition)|author=D. Khattar|publisher=Pearson Education India|page=10.8|isbn=8-131-728-765|url=http://books.google.co.uk/books?id=5OffscY1FGYC&pg=SA10-PA8&dq=Arithmetico-geometric+sequence&hl=en&sa=X&ei=hY8PUo2qM7HZ0QWPwoDgBw&redir_esc=y#v=onepage&q=Arithmetico-geometric%20sequence&f=false}}

*{{cite book|title=Comprehensive Mathematics XI|author=P. Gupta|publisher=Laxmi Publications|page=380|isbn=8-170-085-977|url=http://books.google.co.uk/books?id=rQtlgvS598MC&pg=PA380&dq=Arithmetico-geometric+sequence&hl=en&sa=X&ei=hY8PUo2qM7HZ0QWPwoDgBw&redir_esc=y#v=onepage&q=Arithmetico-geometric%20sequence&f=false}}

[[Category:Mathematical series]]

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en>John of Reading: Typo/general fixing, replaced: of had → of them had using AWB