A flow diagram takes an input n and produces an output Tn. Given the following input-output pairs, find the rule for the sequence: (5, 17), (10, 32), (15, 47), (1, 5).

The given input-output pairs are:

• Input ($n$) = 5, Output ($T_n$) = 17
• Input ($n$) = 10, Output ($T_n$) = 32
• Input ($n$) = 15, Output ($T_n$) = 47
• Input ($n$) = 1, Output ($T_n$) = 5

Assume the rule is linear, of the form $T_n = an + b$.
Calculate the change in output for a given change in input:
For inputs 5 and 10:
Change in input: $10 - 5 = 5$
Change in output: $32 - 17 = 15$
The coefficient $a$ is $\frac{\text{change in output}}{\text{change in input}} = \frac{15}{5} = 3$.
So, the rule is $T_n = 3n + b$.

Substitute one of the input-output pairs into the rule $T_n = 3n + b$. Using $n=1$ and $T_n=5$:

The rule is $T_n = 3n + 2$.

For $n=5$: $T_5 = 3(5) + 2 = 15 + 2 = 17$. (Matches)
For $n=10$: $T_{10} = 3(10) + 2 = 30 + 2 = 32$. (Matches)
For $n=15$: $T_{15} = 3(15) + 2 = 45 + 2 = 47$. (Matches)
The rule is consistent for all given pairs.