Find the first five elements of the geometric sequence if a1- a5=-12 and a2 +a4=8.

Solution:
1. Given:
- Sum of the first and fifth terms: a_1 + a_5 = -12
- Sum of the second and fourth terms: a_2 + a_4 = 8

2. Express the terms of the geometric sequence:
- a_2 = a_1r
- a_4 = a_1r^3
- a_5 = a_1r^4

3. Use the given conditions:
- a_1 + a_1r^4 = -12
- a_1r + a_1r^3 = 8

4. Factor out a_1 in both equations:
- a_1(1 + r^4) = -12
- a_1r(1 + r^2) = 8

5. Solve the second equation for a_1:
- a_1 = \frac{8}{r(1 + r^2)}

6. Substitute a_1 into the first equation:
- \frac{8}{r(1 + r^2)}(1 + r^4) = -12

7. Simplify the equation:
- \frac{8(1 + r^4)}{r(1 + r^2)} = -12
- 8(1 + r^4) = -12r(1 + r^2)
- 8 + 8r^4 = -12r - 12r^3
- 8r^4 + 12r^3 + 12r + 8 = 0

8. Factor the polynomial:
- r^3(2r + 3)(4r + 4) = 0
- Possible values for r are solutions to 2r + 3 = 0 or 4r + 4 = 0.

9. Solve for r:
- r = -\frac{3}{2} or r = -1

10. Check the solutions:
a. For r = -1:
- Substitute into the first equation:
- a_1(1 + (-1)^4) = -12
- 2a_1 = -12
- a_1 = -6

b. Find a_2, a_3, a_4, a_5:
- a_2 = a_1r = -6 \cdot (-1) = 6
- a_3 = a_1r^2 = -6 \cdot 1 = -6
- a_4 = a_1r^3 = -6 \cdot (-1) = 6
- a_5 = a_1r^4 = -6 \cdot 1 = -6

11. The first five terms of the geometric sequence are:
- a_1 = -6
- a_2 = 6
- a_3 = -6
- a_4 = 6
- a_5 = -6