Study the given table. Complete it where the items are missing. Describe the relationship between the input and output values.

Question 1: Study the given table. Complete it where the items are missing. Describe the relationship between the input and output values.

• Step 1: Determine the rule for the table. Let the input be $n$ and the output be $m$. From the table, we observe the pairs $(1,3)$, $(2,5)$, $(3,7)$, $(4,9)$. The output increases by 2 for every increase of 1 in the input. This indicates a linear relationship of the form $m = An + B$. The rate of change $A = \frac{5-3}{2-1} = 2$. Substitute $A=2$ and an input-output pair, for example $(1,3)$, into the equation:

  $$3 = 2(1) + B$$ $$3 = 2 + B$$ $$B = 1$$

  The rule for the relationship is $m = 2n + 1$.

• Step 2: Complete the missing values in the table. Using the rule $m = 2n + 1$: For the input after 4, which is 5:

  $$m = 2(5) + 1 = 10 + 1 = 11$$

  For input $n=10$:

  $$p = 2(10) + 1 = 20 + 1 = 21$$

  For output $m=23$:

  $$23 = 2z + 1$$ $$22 = 2z$$ $$z = 11$$

  The completed table is:

  $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Input } (n) & 1 & 2 & 3 & 4 & \mathbf{5} & 10 & \mathbf{11} \\ \hline \text{Output } (m) & 3 & 5 & 7 & 9 & \mathbf{11} & \mathbf{21} & 23 \\ \hline \end{array}$$

• Step 3: Describe the relationship.

  • Words: The output value is obtained by multiplying the input value by two and then adding one.
  • Number sentence: $m = 2n + 1$
  • Flow diagram: Input ($n$) $\rightarrow$ Multiply by 2 $\rightarrow$ Add 1 $\rightarrow$ Output ($m$)

Question 2: Use the given rule to complete the flow diagram and answer related questions.

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