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

1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 1 9 1 9 36 84 126 126 84 36 9 1 10 1 10 45 120 210 252 210 120 45 10 1

      In order to exactly prove Eq. (2.236), one should start by realizing that

(2.249)

      arises when one sets n = 0; definition of power of nil exponent and summation, as well as lead to

(2.250)

      that entails a universal condition. Suppose now that Eq. (2.236) is valid for a given n; its left‐hand side would then read

      (2.251)

      for n + 1, where power splitting and application of the distributive property meanwhile took place; insertion of Eq. (2.236) leads to

(2.252)

since it applies to (x + y)ⁿ by hypothesis. After factoring x and y, Eq. (2.252) becomes

      (2.253)

      where the last term of the first summation and the first term of the second summation may to advantage be made explicit as

(2.254)

Equation (2.254) may take the simpler form

(2.255)

because y⁰ = x⁰ = 1 and . Replacement of the counting variable k + 1 in the first summation of Eq. (2.255) by k permits transformation to

      (2.256)

      in view of the similarity of lower and upper limits for the two summations, one may lump them to get

      (2.257)

      – where x^k y^n+1−k may, in turn, be factored out as

(2.258)

Equation (2.248) may now be invoked to reformulate Eq. (2.258) to

      (2.259)

      while the first and last terms may be rewritten to get

      (2.260)

      association of such terms to the outstanding summation is then fully justified, viz.

(2.261)

If Eq.