# Strategy 7: Find a Pattern

Patterns are found throughout mathematics. Identifying a pattern is often the first step in understanding a complex mathematical situation. Pattern recognition yields insight that may point in the direction of a complete solution to the problem or simply help you generate a hypothesis, which requires further exploration using another strategy. In a problem where you suspect there is a pattern but don’t recognize it yet, working with particular instances can help you identify the pattern. Once a pattern is identified, it can be used to answer questions.

• This strategy is used in the following sample question.

This is a Multiple-Choice – Select One or More Answer Choices question.

1. Which of the following could be the units digit of where n is a positive integer?

Indicate all such digits.

 (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 (F) 5 (G) 6 (H) 7 (I) 8 (J) 9

Explanation

The units digit of is the same as the units digit of for all positive integers n. To see why this is true for compute by hand and observe how its units digit results from the units digit of Because this is true for every positive integer n, you need to consider only powers of 7. Beginning with and proceeding consecutively, the units digits of 7,   and are 7, 9, 3, 1, and 7, respectively. In this sequence, the first digit, 7, appears again, and the pattern of four digits, 7, 9, 3, 1, repeats without end. Hence, these four digits are the only possible units digits of and therefore of The correct answer consists of Choices B (1), D (3), H (7), and J (9).

Note: This question and explanation also appear as an example of Strategy 12. 