Final answer:
To solve the recurrence relation using iterative substitution, we substitute the previous term in the recurrence relation to find the next term. This process continues until we reach the desired solution. The asymptotic solution depends on the type of terms in the recurrence relation: exponential, polynomial, logarithmic, or factorial.
Step-by-step explanation:
To solve the recurrence relation using iterative substitution, we substitute the previous term in the recurrence relation to find the next term. This process continues until we reach the desired solution. To find the asymptotic solution, we consider the behavior of the terms as n approaches infinity.
(a) Exponential: If the recurrence relation involves exponential terms, the solution will be of the form a^n, where a is a constant. For example, if the recurrence relation is T(n) = 2T(n - 1), the solution will be T(n) = 2^n.
(b) Polynomial: If the recurrence relation involves polynomial terms, the solution will be of the form n^k, where k is a constant. For example, if the recurrence relation is T(n) = nT(n - 1), the solution will be T(n) = n! (factorial).
(c) Logarithmic: If the recurrence relation involves logarithmic terms, the solution will be of the form log(n). For example, if the recurrence relation is T(n) = T(n/2) + 1, the solution will be T(n) = log(n).
(d) Factorial: If the recurrence relation involves factorial terms, the solution will be of the form n!. For example, if the recurrence relation is T(n) = nT(n - 1), the solution will be T(n) = n!.