as a function of n – for selected values of ratio, k.
An alternative proof of Eq. (2.93) comes from finite induction; one should first confirm that it applies to n = 0, i.e.
(2.100)
Furthermore, one should prove that, if Eq. (2.93) is valid for any given n, then it will necessarily apply to n + 1, viz.
according to Eq. (2.87), one has it that
(2.102)
per definition – or, after splitting the summation,
Elimination of parenthesis transforms Eq. (2.103) to
where Eq. (2.87), coupled with validity of Eq. (2.93) for a given n permit transformation of Eq. (2.104) to
once u0 is factored out, Eq. (2.105) becomes
(2.106)
and elimination of parenthesis produces, in turn,
Cancelation of symmetrical terms, complemented with condensation of factors alike transform Eq. (2.107) to
(2.108)
that coincides with Eq. (2.101); therefore, Eq. (2.93) will be valid for n + 1 provided that it holds for n – and one accordingly concludes, together with its validity for n = 0, that Eq. (2.93) is universally applicable to (every) n.
2.1.3 Arithmetic/Geometric Series
An arithmetic/geometric sequence is the result of multiplying a geometric progression by an arithmetic progression, and looks like
in general – with k1 serving as increment and k2 serving as ratio, besides u0 serving as first term; one may write Eq. (2.109) in a more condensed form as
– which is a hybrid of Eqs. (2.76) and (2.87). If both sides are multiplied by k2, then Eq. (2.109) becomes
so ordered subtraction of Eq. (2.111) from Eq. (2.109) is in order to yield
– where parentheses were taken out for convenience; after factoring
One may now factor out k1 k2 or
(2.114)
where the content of the first parenthesis in the right‐hand side is but a geometric series containing n terms; insertion of Eq. (2.93) with u0 = 1 unfolds
(2.115)
while division of both sides by 1 − k2 gives then rise to
A graphical representation of Eq.
