Question

Solve the following: a) Give a recursive definition of the function f for both/equations with n∈Z

+
.
f(n)=2
n

f(n)=5n+2

b) Give a recursive definition of the sequence {a
n

} for both equations with n∈Z
+
. a
n

=2
n
a
n

=3n−5 c) Give a recursive definition with initial conditions of the set S for both sets.
{3,7,11,15,19,23,…}
{…−5,−3,−1,1,3,5,…}

Answer 1

Answer and Explanation:

a) Recursive definition of the function f for both equations with n∈Z:For the first equation, f(n) = 2^n, the recursive definition is:

f(0) = 1 (base case)

f(n) = 2 * f(n - 1) for n > 0For the second equation, f(n) = 5n + 2, the recursive definition is:

f(0) = 2 (base case)

f(n) = f(n - 1) + 5 for n > 0b) Recursive definition of the sequence {a_n} for both equations with n∈Z:For the first equation, a_n = 2^n, the recursive definition is:

a_0 = 1 (base case)

a_n = 2 * a_(n - 1) for n > 0For the second equation, a_n = 3n - 5, the recursive definition is:

a_0 = -5 (base case)

a_n = a_(n - 1) + 3 for n > 0c) Recursive definition with initial conditions of the set S for both sets:For the set {3, 7, 11, 15, 19, 23, ...}:

S_1 = 3 (initial condition)

S_n = S_(n - 1) + 4 for n > 1For the set {... -5, -3, -1, 1, 3, 5, ...}:

S_1 = -5 (initial condition)

S_n = S_(n - 1) + 2 for n > 1