Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier Malcata

      Furthermore, one should prove that, if Eq. (2.93) is valid for any given n, then it will necessarily apply to n + 1, viz.

(2.101)

      according to Eq. (2.87), one has it that

      (2.102)

      per definition – or, after splitting the summation,

(2.103)

Elimination of parenthesis transforms Eq. (2.103) to

(2.104)

where Eq. (2.87), coupled with validity of Eq. (2.93) for a given n permit transformation of Eq. (2.104) to

(2.105)

once u₀ is factored out, Eq. (2.105) becomes

      (2.106)

and elimination of parenthesis produces, in turn,

(2.107)

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

(2.109)

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

(2.110)

      – which is a hybrid of Eqs. (2.76) and (2.87). If both sides are multiplied by k₂, then Eq. (2.109) becomes

(2.111)

so ordered subtraction of Eq. (2.111) from Eq. (2.109) is in order to yield

(2.112)

– where parentheses were taken out for convenience; after factoring out in the left‐hand side, and condensing similar terms in the right‐hand side, Eq. (2.112) becomes

(2.113)

One may now factor out k₁ k₂ or (as appropriate) in Eq. (2.113) to obtain

      (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

(2.116)

A graphical representation of Eq. (2.116) is made available in Fig. 2.6 – after recalling Eq. (2.72) and dividing both sides by u₀, i.e.

(2.117)

      n, k₁/u₀ and k₂ were consequently utilized as independent parameters. An increasing n systematically produces a higher‐value series – and a similar