The Explicit Rule of Arithmetic Sequences

Every arithmetic sequence has a common difference d. Therefore, it is possible to obtain every term of the sequence by adding the common difference to the first term a_1 an appropriate number of times.

Tables can help in identifying the pattern and writing a general expression.

n	a_n	Rewrite
1	a_1	a_1 + 0 * d
2	a_2	a_1 + 1 * d
3	a_3	a_1 + 2 * d
4	a_4	a_1 + 3 * d
5	a_5	a_1 + 4 * d

The coefficient of the common difference is always 1 less than the value of the position n. This makes it possible to write an explicit rule like the following formula.

A sequence can be thought of as a set of coordinate pairs where the first coordinate is the position n and the second coordinate is the term value a_n. (1,a_1), (2,a_2), (3,a_3), ... As the position increases by 1, the value of the term increases, or decreases, by a constant. Therefore, the rate of change between two consecutive coordinate pairs is constant and equal to d. That means an arithmetic sequence is a linear function with a slope d.

Therefore, the explicit rule for the sequence can be written by substituting the coordinate pair ( 1, a_1) into the point-slope form of a line. Point-Slope Form y - y_1 = m(x - x_1) [0.8em] Explicit Rule a_n - a_1 = d(n- 1) Finally, the explicit rule can be rewritten to the form given at the beginning of this proof. a_n - a_1 = d(n-1) ⇕ a_n = a_1 + (n-1)d