Answer:
see explanation
Step-by-step explanation:
The n th term of an arithmetic progression is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
Given [tex]a_{6}[/tex] = 12 and [tex]a_{8}[/tex] = 22, then
a₁ + 5d = 12 → (1)
a₁ + 7d = 22 → (2)
Subtract (1) from (2) term by term to eliminate a₁
2d = 10 ( divide both sides by 2 )
d = 5
Substitute d = 5 into (1) to find a₁
a₁ + 5(5) = 12
a₁ + 25 = 12 ( subtract 25 from both sides )
a₁ = - 13
Thus
[tex]a_{2}[/tex] = - 13 + 5 = - 8
[tex]a_{n}[/tex] = - 13 + 5(n - 1) = - 13 + 5n - 5 = 5n - 18 ← n th term
