In Example 12.1, suppose the initial plants are fertilized with genotype AA, the first generation is fertilized with genotype Aa, the second generation is fertilized with genotype AA, and this alternating pattern of fertilization is kept up. Find formulas for the fractions of the plants of genotypes AA, Aa, and aa in the n-th generation.

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EXAMPLE 12. Suppose a farmer has a large population of plants consisting of some distribution of all three possible genotypes AA, Aa, and aa. He desires to undertake a breeding program in which each plant in the population is always fertilized with a plant of genotype AA. We want to derive an expression for the distribution of the three possible genotypes in the population after any number of generations. For n = 0, 1, 2, ..., let us set a = fraction of plants of genotype AA in n-th generation, b = fraction of plants of genotype Aa in n-th generation, c = fraction of plants of genotype aa in n-th generation. Thus, a, b , and c specify the initial distribution of the geno- types. We also have that a +b +c=1 for n= 0, 1, 2,... From Table 12. 1, we may determine the genotype distribution of each generation from the genotype distribution of the preceeding genera- tion by the following equations: an = an-1 +20n-1 by = Cy_1+ 20n-1 n = 1, 2,... (12. ] Cy =0. For example, the first of these three equations states that all the offspring of a plant of genotype AA will be of genotype AA under this breeding program, and half of the offspring of a plant of geno type Aa will be of genotype AA. Equations (12.1) can be written in matrix notation as x (") = MY (n-1) n = 1, 2, ... (12. 2 where 'n x (12 ) b x (n- 1 ) an-1 0 n- 1 and M= 0 1 Cn-1 0 0