Find the nth term and the 12th term of the list of numbers : 5, 2, -1, -4, …

The given list of numbers is 5, 2, -1, -4, ….

Here, a₂ - a₁ = 2 - 5 = -3,     a₃ - a₂ = -1 - 2 = -3,

          a₄ - a₃ = -4 - (-1) = -3

i.e. any term - preceding term = -3, a fixed number.
So, the given list of numbers forms an A.P. with a = 5 and d = -3.

∴ nth term = a_n = a + (n - 1)d = 5 + (n - 1)(-3) = 5 - 3n + 3 = 8 - 3n.

Putting n = 12, we get
a₁₂ = 8 - 3(12) = 8 - 36 = -28.

Hence, a₁₂ = -28 and a_n = 8 - 3n.