Write the closed form of the ordinary generating function fo

Write the closed form of the ordinary generating function for each of the following sequences.

(a) 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, . . .

(b) 0, 0, 1, 3, 9, 27, 81, 243, . . .

Solution

(a) Let {x_n} be the sequence {2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0 . . .}

Note that x_n = 2 for all n = 1, 5, 9, 13... ; and x_n = 0 for all n N – {1, 5, 9, 13........}

In other words, x_n = 2 for all n of the form 1 + 4k, where k = 0, 1, 2, 3 … and x_n = 0 for all n N – {1 + 4k : k = 0, 1, 2, 3 …}

(b) Let {y_n} be the sequence {0, 0, 1, 3, 9, 27, 81, 243 . . .}

Hence, y₁ = 0, y₂ = 0, y₃ = 1, y₄ = 3, y₅ = 9, y₆ = 27, y₇ = 81, y₈ = 243,…

y₁ = 0, y₂ = 0, y₃ = 1 = 3⁰, y₄ = 3 = 3¹, y₅ = 9 = 3², y₆ = 27 = 3³, y₇ = 81 = 3⁴, y₈ = 243 = 3⁵,……

y₃ = 3^3-3, y₄ = 3^4-3, y₅ = 3^5-3, y₆ = 3^6-3, y₇ = 3^7-3, y₈ = 3^8-3,……..

Hence, we can write a compact form as y_n = 3^n-3 for all n 3

Therefore, the closed form of the sequence {y_n} is y₁ = 0, y₂ = 0, y_n = 3^n-3 for all n 3         (Answer)