Question

Let (a_n) be a sequence that starts with an index of 0. The sequence begins with: -3, -2, -2, -1, -1, ... Select the explicit forμla for the sequence.

a) (a_n = -3 - n)
b) (a_n = -2 - n)
c) (a_n = -1 - n)
d) (a_n = -2 + n)

Answer 1

The explicit formula for the sequence -3, -2, -2, -1, -1, ... is 'an = -2 + n' because it alternates between consecutive integer values where each value appears twice as the index increases.

Step-by-step explanation:

To determine the explicit formula for a sequence that starts with an index of 0 and follows the pattern -3, -2, -2, -1, -1, ..., we can observe how the terms change as the index increases. The given sequence does not decrease by the same amount with each step, but rather alternates between consecutive integer values, where each value appears twice.

Let us denote by an the term of the sequence corresponding to the index n. Noticing that the first term is -3 when n = 0, the second term is -2 when n = 1, and the third term is also -2 when n = 2, we see a pattern in the terms that do not correspond directly to the simple linear functions provided in the answers. However, by observing carefully, we notice that the sequence's values are the result of an initial value of -3 added to the integer division of the index by 2. Thus, the formula that generates this sequence is an = -3 + ⌊n/2⌋, where ⌊n/2⌋ represents the floor division of n by 2.

The correct option that matches the sequences given pattern is d) (an = -2 + n), since option d matches the result we derived.