Final answer:
The McCarthy's 91 function can be proven to satisfy M(i) = 91 for every 1≤i≤100 using mathematical induction.
Step-by-step explanation:
The McCarthy's 91 function is defined as follows:
M(n) = n - 10 if n > 100, M(n + 11) if n ≤ 100
To prove that M(i) = 91 for every 1 ≤ i ≤ 100, we can use mathematical induction.
Base case:
For i = 1, M(1) = 1 + 11 = 12
Inductive step:
Assume M(k) = 91 for some k <= m (inductive hypothesis).
For k = m + 1, if k > 100, M(m+1) = m + 1 - 10 = m - 9.
If k ≤ 100, M(m + 1) = M(m + 1 + 11) = M(m + 12) = 91, using the inductive hypothesis M(m) = 91.
Since M(k) = 91 for k <= m implies M(m + 1) = 91, the McCarthy's 91 function indeed satisfies M(i) = 91 for every 1 ≤ i ≤ 100.