Determine the general term of the number pattern above. How many dots will be needed to build the 10th figure? 110 dots are needed for the kth figure. Calculate the value of k.

• 1.1 What is a linear number pattern? A linear number pattern, also known as an arithmetic sequence, is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference.

• 1.2 Explain what a general term (or $n^{th}$ term) of a sequence is. The general term (or $n^{th}$ term) of a sequence is a formula, usually denoted as $T_n$ or $a_n$, that allows you to calculate any term in the sequence by substituting its position number $n$.

• 1.3 What is the difference between a term number and a term value? The term number refers to the position of a term in a sequence (e.g., 1st, 2nd, 3rd term), while the term value is the actual numerical value of the term at that specific position.

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Question 2:

Step 1: Identify the number of dots in each figure and determine the common difference.
Figure 1 ($T_1$) has 6 dots.
Figure 2 ($T_2$) has 9 dots.
Figure 3 ($T_3$) has 12 dots.
The common difference $d$ is:

$$d = T_2 - T_1 = 9 - 6 = 3$$ $$d = T_3 - T_2 = 12 - 9 = 3$$

The common difference is 3.

• 2.1 Determine the general term of the number pattern above. The formula for the general term of a linear pattern is $T_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. Substitute $a=6$ and $d=3$:

  $$T_n = 6 + (n-1)3$$ $$T_n = 6 + 3n - 3$$ $$T_n = 3n + 3$$

  The general term is $\boxed{\mathbf{T_n = 3n + 3}}$.

• 2.2 How many dots will be needed to build the 10th figure? To find the number of dots for the 10th figure, substitute $n=10$ into the general term $T_n = 3n + 3$:

  $$T_{10} = 3(10) + 3$$ $$T_{10} = 30 + 3$$ $$T_{10} = 33$$

  The 10th figure will need $\boxed{\mathbf{33 \text{ dots}}}$.

• 2.3 116 dots are needed for the $k^{th}$ figure. Calculate the value of $k$. Set the general term $T_k$ equal to 116 and solve for $k$:

  $$3k + 3 = 116$$

  Subtract 3 from both sides:

  $$3k = 116 - 3$$ $$3k = 113$$

  Divide by 3:

  $$k = \frac{113}{3}$$ $$k \approx 37.67$$

  The value of $k$ is $\boxed{\mathbf{\frac{113}{3}}}$. Since $k$ represents a figure number, it is typically expected to be a positive integer. As $\frac{113}{3}$ is not an integer, 116 dots cannot form an exact figure in this pattern.