Answer:
d = 5 and a₁ = - 4
Step-by-step explanation:
The n th term of an arithmetic sequence is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
Given a₃ = 6, then
a₁ + 2d = 6 → (1)
The sum to n terms of an arithmetic sequence is
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] [ 2a₁ + (n - 1)d ]
Given [tex]S_{12}[/tex] = 282, then
6 [ 2a₁ + 11d ] = 282 ( divide both sides by 6 )
2a₁ + 11d = 47 → (2)
We can now solve (1) and (2) for d and a₁
Multiply (1) by - 2
- 2a₁ - 4d = - 12 → (3)
Add (2) and (3) term by term
7d = 35 ( divide both sides by 7 )
d = 5
Substitute d = 5 in (1) and solve for a₁
a₁ + 10 = 6 ( subtract 10 from both sides )
a₁ = - 4
