Finding the n^th Term of an Arithmetic Sequence

Example 1:

Find the 27^th term of the arithmetic sequence 5, 8, 11, 14,···.

a₁ = 5, d = 8 – 5 = 3

Example 2:

Find the 40^th term for the arithmetic sequence in which
a₈ = 60 and a₁₂ = 48.

Substitute 60 for a₈ and 48 for a₁₂ in the formula
a_n = a₁ + (n – 1)d  to obtain a system of linear equations in terms of a₁ and d.

Subtract the second equation from the first equation and solve for d.

12 = –4d
 –3 = d

Then 60 = a₁ + 7(–3).  Solve for a.
        60 = a₁– 21     
        81 = a₁

Now use the formula to find a₄₀.

a₄₀ = 81 + 39(–3) = 81 – 117 = –36.