This is Discrete Math Give recursive definitations of each o

This is Discrete Math.

Give recursive definitations of each of the following sequences:

a. 1, 5, 5², 5³, 5⁴,...

b. 5, 3, 1, -1, -3,...

Solution

(a) Given the sequence (1, 5, 5², 5³, …),

S₀ = 1, s₁= 5.

The common ratio between a term and its previous term is always 5 leaving first term.

Thus,

s_n = 5s_n-1   for n greater than and equal to 1.

and for n = 0, s₀ = 1.

(b) Given the sequence (5, 3, 1, -1, -3, …),

S₀ = 5, s₁= 3.

The difference between a term and its previous term is always 2,

Thus,

s_n = s_n-1 - 2 for for n greater than and equal to 1.