Remarks about closed formula for sequence

Remarks.

1. The sequence in Example \(1\) is given by a formula in terms of \(n\) only. Such a formula is said to be a closed formula in \(n\). Humans love to define sequences using closed formulae since the value of the \(n^{\mathrm{th}}\) term can be calculated directly by substituting the desired position \(n\).
2. The sequence in Example \(2\) is defined using a recurrence relationship among \(u_n\) and some of the previous terms \(u_1, u_2, \dots, u_{n-1}\). Humans find this method very tedious because one needs to compute all of \(u_1, u_2, \dots, u_{n-1}\) in order to obtain \(u_n\). Thus, we delegate such mundane and troublesome computational tasks to the computers (e.g., spreadsheets or GDC tables).
3. Even though the closed formula for generating a sequence is unavailable, one can still rely on the computational prowess of computers to perform this kind of calculations. Example \(3\)(ii) shows how to harness the computing power of GDC in obtaining by brute force the value of \(n\) for which \(u_n\) first exceeds \(256\) even when the closed formula for \(u_n\) is not available.