Arithmetic formulas originate from the need to determine the value or position of a specific term within an arithmetic sequence, where the difference between successive terms is a constant d, such as "an = a1 - (n - 1)d." Geometric formulas are derived from a similar need but applied to a geometric sequence with a common ratio of r, such as "an = a1 * r^(n - 1)."
Other important arithmetic and geometric formulas include the formulas for finding the partial sums of arithmetic and geometric sequences. Given a sequence of figures, the student can apply either a geometric or arithmetic formula to find a specific missing term in the series or the sum of the series. The first step is to determine whether the sequence is a geometric or arithmetic sequence.
For an arithmetic sequence, the student can easily determine the common difference by subtracting the successive terms in the sequence. However, for a geometric sequence, the student must pick a pair of numbers from any position in the sequence, and the numbers do not have to be successive terms. Students then divide the terms to obtain the common ratio. Having obtained the common ratio, the student can find any term through simple multiplication.