Finding the Unknown | FIO

Question 24

Two separate figures are given below. Each figure shows the first few positions in a sequence of arrangements made with sticks. Identify the pattern and answer the following questions for each figure:

(a) How many squares are in position number 11 of the sequence? (b) How many sticks are needed to make the arrangement in position number 11 of the sequence? (c) Can an arrangement in this sequence be made using exactly 85 sticks? If yes, which position number will it correspond to? (d) Can an arrangement in this sequence be made using exactly 150 sticks? If yes, which position number will it correspond to?

Solution

Understand the Question

First Figure (Squares): The number of squares increases by $3$ with each position. For position $n$ , the rule is: $S_n = 1 + 3n$
Second Figure (Sticks): The number of sticks starts at $13$ for position $1$ and adds $9$ sticks for each subsequent position. For position $n$ , the rule is: $T_n = 13 + 9(n - 1)$
To determine if a specific number of sticks can form an arrangement, solve for $n$ . If $n$ is a positive whole number, the arrangement is possible; otherwise, it is not.

(a) How many squares are in position number 11 of the sequence?

Step 1 · Calculate the Number of Squares for Position 11

Observing the pattern for the number of squares in position $n$ :

Position 1: $1 + 1 \times 3 = 4$
Position 2: $1 + 2 \times 3 = 7$
Position 3: $1 + 3 \times 3 = 10$

The general formula for the number of squares is: $S_n = 1 + 3n$

Substitute $n = 11$ :

\begin{aligned} S_{11} &= 1 + 3 \times 11 \\[0.6em] &= 1 + 33 \\[0.6em] &= 34 \end{aligned}

Answer

(a) 34 squares

(b) How many sticks are needed to make the arrangement in position number 11 of the sequence?

Step 1 · Calculate the Number of Sticks for Position 11

Observing the pattern for the number of sticks in position $n$ :

Position 1: $13 + 9 \times (1 - 1) = 13$
Position 2: $13 + 9 \times (2 - 1) = 22$
Position 3: $13 + 9 \times (3 - 1) = 31$

The general formula for the number of sticks is: $T_n = 13 + 9(n - 1)$

Substitute $n = 11$ :

\begin{aligned} T_{11} &= 13 + 9 \times (11 - 1) \\[0.6em] &= 13 + 9 \times 10 \\[0.6em] &= 13 + 90 \\[0.6em] &= 103 \end{aligned}

Answer

(b) 103 sticks

(c) Can an arrangement in this sequence be made using exactly 85 sticks? If yes, which position number will it correspond to?

Step 1 · Solve for Position Number with 85 Sticks

Set the formula for the number of sticks equal to $85$ : $13 + 9(n - 1) = 85$

\begin{aligned} 9(n - 1) &= 85 - 13 \\[0.6em] 9(n - 1) &= 72 \\[0.6em] n - 1 &= \dfrac{72}{9} \\[0.6em] n - 1 &= 8 \\[0.6em] n &= 8 + 1 \\[0.6em] n &= 9 \end{aligned}

Since $n = 9$ is a positive integer, an arrangement can be made.

Answer

(c) Yes, it corresponds to position number 9.

(d) Can an arrangement in this sequence be made using exactly 150 sticks? If yes, which position number will it correspond to?

Step 1 · Solve for Position Number with 150 Sticks

Set the formula for the number of sticks equal to $150$ : $13 + 9(n - 1) = 150$

\begin{aligned} 9(n - 1) &= 150 - 13 \\[0.6em] 9(n - 1) &= 137 \\[0.6em] n - 1 &= \dfrac{137}{9} \\[0.6em] n - 1 &= 15.222\dots \\[0.6em] n &= 16.222\dots \end{aligned}

Since $n$ is not a whole number, no arrangement can be made with exactly $150$ sticks.

Answer

(d) No

Common Mistakes

Index Off-by-One Error: Using $13 + 9n$ instead of $13 + 9(n - 1)$ for the number of sticks in position $n$ .
Non-Integer Positions: Forgetting that position numbers must be positive integers ( $n \in \mathbb{N}$ ). A fractional value of $n$ means that exact number of sticks cannot form a complete figure in the sequence.

Question 24

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