2. Representing Patterns (A)
  3. Lesson 9: Representing Sequences Using Equations

Exercises


  1. Consider the following sequence of images.
    Term 1 has 5 squares, term 2 has 8 squares, term 3 has 11 squares, and term 4 has 14 squares.
    1. Write an equation to represent the number of squares, \(S\), in term \(n\) of the sequence.
    2. How many squares are needed to construct the \(20^{th}\) term in the sequence?
  2. Consider the following sequence of images where the first term contains two adjoining cubes, and one cube is added each time to obtain the next image. Each exposed face of the cubes is painted.
    Term 1 contains 2 squares with 1 adjoining side, term 2 contains 3 squares with 2 adjoining sides, term 3 contains 4 squares with 3 adjoining sides and term 4 contains 5 squares with 3 adjoining sides.
    1. Write an equation to represent the number of painted faces, \(P\), with respect to the number of blocks, \(b\), in an image.
    2. How many blocks are required to form the image having \(70\) painted faces?
  3. Consider each of the sequences given below. If the difference between consecutive terms remains constant, then in which of the following sequences does the integer \(512\) appear?
    1. \(8,~15,~22,~29,\ldots\)
    2. \(23, ~26,~ 29,~32, \ldots\)
    3. \(12,~ 24,~ 36,~ 48, \ldots\)
  4. The printing company charges an upfront cost of \($150\) for a printing job and then a fee of \($7\) per book printed.  You are using this company to print yearbooks for your school.
    1. What is an equation for the total cost, \(C\), if \(y\) yearbooks are printed?
    2. How much will it cost for \(100\) yearbooks to be printed?
    3. If your budget is \($1900\), then how many yearbooks can you get printed?
  5. A scuba diver dives at a constant rate of \(10\) metres per minute. Ashley used the equation \(D=10t\) to describe the depth, \(D\), of the scuba diver after \(t\) minutes.
    1. Explain why \(D\) and \(t\) are good choices to use as variables in the equation.
    2. Did Ashley have to choose the letters \(D\) and \(t\) to use as variables? What other letters could she have chosen?
    3. What letters might you avoid using as variables in this problem? Explain why.
  6. The following table represents what Jason knows about a sequence of numbers.
    Term Number Term Value
    \(1\) \(6\)
    \(2\) \(13\)
    \(5\) \(34\)
    \(9\) \(62\)
    1. Find the differences between adjacent term values in the table. What do you notice?
    2. Graph the sequence.
    3. How do we find a possible general term for the sequence?
    1. Calculate each sum.
      1. \(1+3+5\)
      2. \(1+3+5+7\)
      3. \(1+3+5+7+9+11+13\)
    2. Did you notice a pattern appearing in the above calculations?  What is the pattern?
    3. Explain how the following diagram helps us to understand the pattern we found in part b).  Can you use this pattern to calculate other, longer, sums of consecutive odd numbers?
      The following terms are arranged to form a capital L. Term 1 contains 1 square, term 2 contains 3 squares, term 3 contains 5 squares and term 4 contains 7 squares.
  8. Let \(S\) represent the sum of the first \(n\) positive integers.  Then, \(S\) and \(n\) satisfy the following equation: \[S = \dfrac{n(n+1)}{2}\]Verify the two equalities below for \(n = 5\) and \(n = 10\):
    \(1+2+3+4+5 =\dfrac{5(5+1)}{2} \quad \) and \(\quad 1+2+3+\cdots+9+10 = \dfrac{10(10+1)}{2}\)
    Calculate each sum below. You can take the time to add up all of the numbers on a calculator if you'd like, but that won't be much fun! A much better option is to try and find a shortcut using the above formula.  
    1. \(1+2+3+ \cdots + 99 +100\)
    2. \(19+20+21+\cdots+89+90\)
    3. \(4+8+12+\cdots+96+100\)
    4. \(3+7+11+\cdots+83+87\)