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Arithmeticogeometric sequence

Calculus  



Specialized 

In mathematics, an arithmeticogeometric sequence is the result of the multiplication of a geometric progression with the corresponding terms of an arithmetic progression.
Contents
Sequence, nth term
The sequence has the nth term^{[1]} defined for n ≥ 1 as:
 <math>[a+(n1)d] r^{n1} </math>
are terms from the arithmetic progression with difference d and initial value a.
Series, sum to n terms
An arithmeticogeometric 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,^{[1]}
 <math>S_n = a + [a + d] r + [a + 2 d] r^2 + \cdots + [a + (n  1) d] r^{n  1}</math>
multiply S_{n} 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 rS_{n} from S_{n},
 <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^{n1}\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 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]}
 <math>\lim_{n \to \infty}S_{n} = \frac{a}{1r}+\frac{rd}{(1r)^2}</math>
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).
See also
References
 ^ ^{a} ^{b} ^{c} K.F. Riley, M.P. Hobson, S.J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. p. 118. ISBN 9780521861533.
Further reading
 D. Khattar. The Pearson Guide to Mathematics for the IITJEE, 2/e (New Edition). Pearson Education India. p. 10.8. ISBN 8131728765.
 P. Gupta. Comprehensive Mathematics XI. Laxmi Publications. p. 380. ISBN 8170085977.