Write out the first five terms of the sequence an=(-2)^n. Determine if the sequence is arithmetic or geometric, and then find the common ratio or difference which option below is accurate

Question

The sequence is arithmetic and the difference is 6
The sequence is arithmetic and the difference is -2
The sequence is geometric and the ratio is -2
The sequence is geometric and the ratio is 2

Answer 1

Let's start by calculating the first five terms of the sequence \( a_n = (-2)^n \):

1. For \( n = 1 \): \( a_1 = (-2)^1 = -2 \)
2. For \( n = 2 \): \( a_2 = (-2)^2 = 4 \)
3. For \( n = 3 \): \( a_3 = (-2)^3 = -8 \)
4. For \( n = 4 \): \( a_4 = (-2)^4 = 16 \)
5. For \( n = 5 \): \( a_5 = (-2)^5 = -32 \)

The first five terms of the sequence are:

• \( a_1 = -2 \)
• \( a_2 = 4 \)
• \( a_3 = -8 \)
• \( a_4 = 16 \)
• \( a_5 = -32 \)

Now let's determine if the sequence is arithmetic or geometric.

1. Arithmetic Sequence: A sequence is arithmetic if the difference between consecutive terms is constant. Let's find the differences:
   • \( a_2 - a_1 = 4 - (-2) = 4 + 2 = 6 \)
   • \( a_3 - a_2 = -8 - 4 = -12 \)
   • \( a_4 - a_3 = 16 - (-8) = 16 + 8 = 24 \)
   • \( a_5 - a_4 = -32 - 16 = -48 \)

Since the differences are not constant, the sequence is not arithmetic.

2. Geometric Sequence: A sequence is geometric if the ratio of consecutive terms is constant. Let's find the ratios:
   • \( \frac{a_2}{a_1} = \frac{4}{-2} = -2 \)
   • \( \frac{a_3}{a_2} = \frac{-8}{4} = -2 \)
   • \( \frac{a_4}{a_3} = \frac{16}{-8} = -2 \)
   • \( \frac{a_5}{a_4} = \frac{-32}{16} = -2 \)

Since the ratios are constant and equal to \(-2\), the sequence is indeed geometric, and the common ratio is \(-2\).

Thus, the correct choice is:

• The sequence is geometric and the ratio is -2.