By Tricia Lobo • Updated August 30, 2022
In the early stages of algebra, students learn to identify both arithmetic and geometric sequences. Recognizing patterns is essential, especially when working with fractions. These patterns may be arithmetic, geometric, or a blend of the two. The key is to remain attentive and systematically examine each fraction.
Ask whether a constant value is added to each fraction to produce the next term. For example, consider the sequence 1/8, 1/4, 3/8, 1/2. By converting all denominators to 8, we see 1/8 → 2/8 → 3/8 → 4/8. The progression adds 1/8 each time, so it is an arithmetic sequence.
Determine whether each fraction is obtained by multiplying the previous one by a fixed factor. Take the sequence 1/16, 1/8, 1/4, 1/2 (or 1/(2^4), 1/(2^3), 1/(2^2), 1/2). Each term is twice its predecessor, revealing a geometric progression.
If neither an arithmetic nor a geometric pattern is apparent, consider combinations of operations, such as reciprocals or simultaneous changes to numerators and denominators. For instance, the sequence 2/3, 6/4, 8/12, 24/16 contains terms that are the reciprocals of 2/3 and 8/12, each achieved by multiplying both numerator and denominator by 2.
