Worksheet: Patterns and Conjecture

In this worksheet, we will practice writing a conjecture that describes the pattern in a sequence and use the conjecture to find the nth term.


Find the tenth term of the sequence 13,13,26,39,65,104,169,.


The figure shows the steps to producing a curve 𝐶. It starts as the boundary of the unit square in Figure (a). In Figure (b), we remove a square quarter of the area of the square in (a). In Figure (c), we add a square quarter of the area that we removed in (b). In Figure (d), we remove a square quarter of the area of the square we added in (c). If we continue to do this indefinitely, we will get the curve 𝐶. We let 𝑅 be the region enclosed by 𝐶. By summing a suitable infinite series, find the area of region 𝑅. Give your answer as a fraction.

  • A56
  • B34
  • C45
  • D15
  • E12


An equilateral triangle has a side length of 14 cm, where another triangle is drawn inside of it by connecting the midpoints of its sides. More interior triangles are to be repeatedly drawn the same way as shown in the figure. Find the sum of the perimeters of the first 6 triangles drawn giving the answer to the nearest integer.


A rectangle whose length is 64 cm and width is 48 cm has its sides bisected. These points are then connected creating a rhombus. The sides of the rhombus are bisected and so on forming the figure below. Find the sum to infinity of the perimeters of the figure.


Consider the following sequence of dots.

Find the number of dots in the next two terms of the sequence.

  • A33, 46
  • B35, 41
  • C45, 70
  • D45, 66
  • E25, 34


Following the sequence of the given figures, how many squares will there be in Figure 77?


In the figure, each square represents one square unit. Letting 𝑥 be the number of columns in the shape and 𝑦 its area measured in square units, write a formula for the area of the shape in terms of 𝑥. Then, calculate the value of 𝑦 when 𝑥=6.

  • A𝑦=3𝑥+3, 𝑦=21
  • B𝑦=𝑥1, 𝑦=35
  • C𝑥=𝑦1, 𝑦=37
  • D𝑦=4𝑥1, 𝑦=23


Consider the following growing pattern, shown for 𝑛=1, 𝑛=2, and 𝑛=3.

Write an expression for the number of dots in the 𝑛th such pattern.

  • A2𝑛+1
  • B𝑛+1
  • C2𝑛1
  • D𝑛+2
  • E𝑛1


Isabella is helping her little sister learn the pairs of numbers that add up to 10. She uses Cuisenaire rods to show her the pairs 1–9, 2–8, and so forth, that all add up to 10. Looking at the pattern she has made with the rods, she realizes that it could help her figure out the sum of the numbers between 1 and 9 without adding up all the numbers.

Write the calculation that Isabella made to find 1+2++9.

  • A10×9
  • B8×93
  • C9×9
  • D10×92
  • E9×92

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