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Arithmetico geometric sequence

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In mathematics, an arithmetico-geometric sequence is the result of the multiplication of a geometric progression with the corresponding terms of an arithmetic progression. The corresponding French term refers to a different concept (sequences of the form u n + 1 = a u n + b ) which is a special case of linear difference equations.

Contents

Sequence, nth term

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

[ a + ( n − 1 ) d ] b r 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

∑ k = 1 n [ a + ( k − 1 ) d ] r k − 1 = a + [ a + d ] r + [ a + 2 d ] r 2 + ⋯ + [ a + ( n − 1 ) d ] r n − 1

and the sum to n terms is equal to:

S n = ∑ k = 1 n [ a + ( k − 1 ) d ] r k − 1 = a − [ a + ( n − 1 ) d ] r n 1 − r + d r ( 1 − r n − 1 ) ( 1 − r ) 2 .

Derivation

Starting from the series,

S n = a + [ a + d ] r + [ a + 2 d ] r 2 + ⋯ + [ a + ( n − 1 ) d ] r n − 1

multiply Sn by r,

r S n = a r + [ a + d ] r 2 + [ a + 2 d ] r 3 + ⋯ + [ a + ( n − 1 ) d ] r n

subtract rSn from Sn,

( 1 − r ) S n = [ a + ( a + d ) r + ( a + 2 d ) r 2 + ⋯ + [ a + ( n − 1 ) d ] r n − 1 ] − [ a r + ( a + d ) r 2 + ( a + 2 d ) r 3 + ⋯ + [ a + ( n − 1 ) d ] r n ] = a + d ( r + r 2 + ⋯ + r n − 1 ) − [ a + ( n − 1 ) d ] r n = a + d r ( 1 − r n − 1 ) 1 − r − [ a + ( n − 1 ) d ] r n

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

lim n → ∞ S n = a 1 − r + r d ( 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

    Arithmetico-geometric sequence Wikipedia
    (Text) CC BY-SA


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