If each of the following sequences is either arithmetic or geometric, continue the decimal patterns below. a. $$ 0.4,2.3,4.2,6.1 $$, b. $$ 5,2.5,1.25,0.625 $$, Fill in the blanks below. a. $$ 0.4,2.3,4.2,6.1,8.0,9.9,11.8 $$ (Type integers or decimals. Simplify your answers.) b. $$ 5,2.5,1.25,0.625, \square . \square . \square $$ (Type integers or decimals. Simplify your answers.)

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Let's analyze and continue each sequence:

### a. Arithmetic Sequence
The given sequence is:
$$ 0.4, \, 2.3, \, 4.2, \, 6.1 $$

**Common Difference (d):**
$$ 2.3 - 0.4 = 1.9 $$
$$ 4.2 - 2.3 = 1.9 $$
$$ 6.1 - 4.2 = 1.9 $$

Since the difference between consecutive terms is constant ($$ d = 1.9 $$), it's an **arithmetic sequence**.

**Continuing the Sequence:**
$$
6.1 + 1.9 = 8.0 \
8.0 + 1.9 = 9.9 \
9.9 + 1.9 = 11.8
$$

**Extended Sequence:**
$$ 0.4, \, 2.3, \, 4.2, \, 6.1, \, 8.0, \, 9.9, \, 11.8 $$

### b. Geometric Sequence
The given sequence is:
$$ 5, \, 2.5, \, 1.25, \, 0.625 $$

**Common Ratio (r):**
$$ \frac{2.5}{5} = 0.5 $$
$$ \frac{1.25}{2.5} = 0.5 $$
$$ \frac{0.625}{1.25} = 0.5 $$

Since the ratio between consecutive terms is constant ($$ r = 0.5 $$), it's a **geometric sequence**.

**Continuing the Sequence:**
$$
0.625 \times 0.5 = 0.3125 \
0.3125 \times 0.5 = 0.15625 \
0.15625 \times 0.5 = 0.078125
$$

**Extended Sequence:**
$$ 5, \, 2.5, \, 1.25, \, 0.625, \, 0.3125, \, 0.15625, \, 0.078125 $$

### **Final Answers:**

**a.** $$ 8.0, \, 9.9, \, 11.8 $$

**b.** $$ 0.3125, \, 0.15625, \, 0.078125 $$
a. $$ 8.0, \, 9.9, \, 11.8 $$ b. $$ 0.3125, \, 0.15625, \, 0.078125 $$

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