-2,4,0,8,8,24,40

The given sequence is -2, 4, 0, 8, 8, 24, 40.

Pattern Explanation:

The sequence follows a recurrence relation:

[

a(n) = a(n-1) + 2 cdot a(n-2)

]

with initial terms:

[

a(1) = -2, quad a(2) = 4

]

Verification:

Closed-Form Formula:

The general term can also be expressed as:

[

a(n) = frac{1}{3} cdot 2^n + frac{8}{3} cdot (-1)^n

]

Key Steps:

1. Recurrence Relation: Each term is the sum of the previous term and twice the term before it.

2. Characteristic Equation: Solving (r^2

3. Closed-Form: Combining the exponential solutions gives the explicit formula.

Answer: The sequence follows the recurrence relation (a(n) = a(n-1) + 2a(n-2)) with initial terms (-2) and (4). The nth term is (boxed{frac{2^n + 8(-1)^n}{3}}).