2. Section 5.2 - 5. Sequences - Core Connections Integrated I, 2013

a We are referred to the sequences from a previous exercise which are shown below.

c . 0, 1, 4, 9, \dots i . 8, 2, 0, 2, 8, 18, \dots

Let's see how each sequence is changing.

c . i . 0 ⟶ + 1 1 ⟶ + 3 4 ⟶ + 5 9 \dots 8 ⟶ - 6 2 ⟶ - 2 0 ⟶ + 2 2 ⟶ + 6 8 ⟶ + 10 18 \dots

We can see that in neither of the sequences can the next term be obtained by adding or subtracting a constant term, or by multiplying or dividing by a constant factor. In both of them the following term is obtained by adding a number, which changes according to a specific rule instead.

b We are referred to the sequence shown below.

e . 1, 1, 2, 3, 5, 8, \dots

Let's see how are the consecutive terms changing in this sequence.

e . 1 ⟶ 0 + 1 1 ⟶ 1 + 1 2 ⟶ 1 + 2 3 ⟶ 2 + 3 5 ⟶ 3 + 5 8 \dots

We can see that this sequence is not arithmetic, as the consecutive term is not obtained by adding a constant number. It is not a geometric sequence either, as the new term is not obtained by multiplying by a constant factor. Further more, the following term depends on the two previous terms and not just on the previous one, like in a recursive sequence.

Exercise 45 Page 264