explicit formula
An explicit formula gives a term directly from its position, with no need to know the terms before it. It is a shortcut that lets you teleport to the n-th term: plug in n, and out comes a_n.
It is just a function of the index n. For an arithmetic sequence the explicit formula is a_n = a_1 + (n − 1)d; for a geometric sequence it is a_n = a_1 · r^(n−1). Want the 100th term? Substitute n = 100 once and evaluate — no climbing required.
This contrasts with a recursive formula, which only tells you how to get the next term from earlier ones. Both can describe the same sequence; the explicit form is more efficient for jumping ahead, while the recursive form is often easier to discover from a pattern.
a_n = 4n − 1 gives 3, 7, 11, 15, ...; the 25th term is a_25 = 4·25 − 1 = 99, found in one step.
One substitution reaches a far-off term directly.