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Question

If a cow of mine has a calf at the age of two years, and supposing she goes on having the like every year, and supposing every one of her young to have a female calf at the age of two years, and afterwards every year likewise, and so on. Now, how many do you suppose would spring from that cow and all her descendents in the space of 25 years? We are to count from the birth of the original cow, and it is obvious that the family can produce only females doing the period stated.

Answer 1

Let's categories the cows by the "state" of life they are in. We assume no cow deaths in 25 years.

A_n -> For any year n, A_n represents the number of adult cows (cows actively giving birth, age 2+)

Y_n -> For any year n, Y_n represents the number of 1 year old cows

N_n -> For any year n, N_n represents the number of newborn cows (cows born that year)

We can use a sort of state machine to find A_n+1, Y_n+1, and N_n+1:

A_n+1 = A_{n +} Y_n (last year's adult cows + "graduated" juvenile cows)

Y_{n+1 =} N_n (the previous year's newborn cows, who are all now 1)

N_n+1 = A_n (each of the previous year's adult cows have had one new baby)

Total cows for year n+1 is A_n+1 + Y_n+1 + N_n+1 = A_{n +} Y_n + N_n + A_n

Note that the total number of cows for year n was A_{n +} Y_n + N_n; T_n+1 = T_n + A_n

Also note that A_n = T_n-1, as new babies are only born from cows that were alive two years ago. Therefore, T_n+1 = T_n + T_n-1.

Given starting value of A₀ = 1 and T₀ = 1, we can just continuously update T (and A/Y/N, if the explanation above was unsatisfying) until year 25. You will find that these values resemble the Fibonacci sequence, which makes sense, as each term in the Fibonacci sequence is the sum of the two preceding terms.