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Define the sequence so that is the number of white squares in Step , and define the sequence so that is the number of black squares in Step .
A three step pattern, black squares surrounded by white squares. “Step 1”, 2 black squares in a row, surrounded by 10 white squares. “Step 2”, a 2 by 3 rectangle made of 6 black squares, surrounded by 14 white squares. “Step 3”, a 3 by 4 rectangle made of 12 black squares, surrounded by 18 white squares.
Some situations can be accurately modeled with geometric sequences, arithmetic sequences, or sequences that are neither geometric nor arithmetic.
For example, here is a pattern of black squares surrounded by white squares, growing in steps.
A three step pattern, rectangles made of small white and black squares. “Step 0”, a 3 by 3 rectangle, 9 squares. The perimeter, 8 white squares. The inside, 1 black square. “Step 1”, a 3 by 5 rectangle, 15 squares. The perimeter, 12 white squares. In the center of the rectangle, a row of 3 squares. From left to right, black, white, black. “Step 2”, a 3 by 7 rectangle, 21 squares. The perimeter, 16 white squares. In the center, a row of 5 squares. From left to right, black, white, black, white, black.
The number of white squares in each step grows (8, 13, 18, . . .), with 5 more white squares each time. Since the same number of squares is added each time, the number of white squares forms an arithmetic sequence. The definition for the term of , where is the number of white squares in step , is for .
Geometric sequences are involved in situations such as population growth and scaling. For example, the sequence of areas we got when we imagined cutting a piece of paper in half at each step in an earlier lesson.
Many situations lead to sequences that are neither geometric nor arithmetic. For example, consider this pattern of dots in which a new row of dots introduced in each step:
A dot pattern. “Step 1” consists of 1 row of 1 dot. "Step 2”, consists of two rows of dots that form a triangle. row 1, 1 dot. row 2, 2 dots. “Step 3”, consists of 3 rows of dots that form a triangle. Row 1, 1 dot. row 2, 2 dots. row 3, 3 dots. “Step 4”, consists of 4 rows of dots that form a triangle. Row 1, 1 dot. row 2, 2 dots. row 3, 3 dots. row 4, 4 dots.
The number of dots in each step grows (1, 3, 6, 10, . . .), but there is no constant being multiplied or added to get from term to term. If we create a graph of this sequence showing the number of dots as a function of the step number, the dots would form neither a linear nor an exponential shape. This sequence is neither geometric nor arithmetic, but it does have a pattern that we can define with an equation.