Question

What is the recurrence relation of the sequence 1, 3, 9, 27?

A) T(n) = 3T(n-1)
B) T(n) = 3T(n-1) + 1
C) T(n) = 3T(n-1) + 3
D) T(n) = 3T(n-1) + 6

Answer 1

The recurrence relation for the sequence 1, 3, 9, 27 is T(n) = 3T(n-1), because each term is simply three times the previous term without any additional constants.

Step-by-step explanation:

The sequence provided is 1, 3, 9, 27. We can determine its recurrence relation by observing the pattern in which each term is derived from the previous one. In this sequence, each term is 3 times the previous term. Hence, if we denote the nth term as T(n), then the relationship for each term T(n) can be defined recursively as:

T(n) = 3 × T(n-1)

This relation indicates that to get the nth term, you multiply the (n-1)th term by 3. No additional terms (like '+1', '+3', or '+6') are needed, as each term is exactly 3 times the preceding term without any addends. Therefore, the correct recurrence relation for this sequence is option A: T(n) = 3T(n-1).