Question

Find the sequence that has the recursive formula aₙ =aₙ−1 +4, a₁ =−9, and the explicit formula aₙ =4n−13.

a) 2,6,10,14,…
b) −9,−5,−1,3,…
c) −13,−9,−5,−1,…
d) −5,−1,3,7,…

Answer 1

The correct sequence matching both the recursive formula aₙ = aₙ−1 + 4, a₁ = −9 and the explicit formula aₙ = 4n − 13 is b) -9, -5, -1, 3, …. This sequence was determined by applying both formulas to the first few terms.

Step-by-step explanation:

To find the sequence that corresponds to the given recursive formula and explicit formula, we start by looking at the first term provided, a1 = -9.

Using the recursive formula an = an-1 + 4, and starting with a1 = -9:

• a2 = a1 + 4 = -9 + 4 = -5
• a3 = a2 + 4 = -5 + 4 = -1
• a4 = a3 + 4 = -1 + 4 = 3

This sequence matches option b) -9, -5, -1, 3, …

To verify with the explicit formula, an = 4n - 13:

• a1 = 4(1) - 13 = -9
• a2 = 4(2) - 13 = -5
• a3 = 4(3) - 13 = -1
• a4 = 4(4) - 13 = 3

Thus, the correct sequence is b) -9, -5, -1, 3, …, which follows both the recursive and explicit formulas given.