To solve the problem, let's first understand the relationship between the points. Since point $T$ is between points $S$ and $U$, the sum of the segments $ST$ and $TU$ should equal the entire segment $SU$:

$SU = ST + TU$

We are given:

$SU = 40n - 40$

$ST = 20n - 2$

$TU = 5n + 7$

Now, we can set up the equation:

$40n - 40 = (20n - 2) + (5n + 7)$

Step 1: Simplify the right side

First, combine like terms on the right side:

$40n - 40 = 20n + 5n - 2 + 7$

$40n - 40 = 25n + 5$

Step 2: Solve for $n$

Subtract $25n$ from both sides to isolate the variable $n$:

$40n - 25n - 40 = 5$

$15n - 40 = 5$

Next, add 40 to both sides:

$15n = 45$

Finally, divide both sides by 15:

$n = 3$

Step 3: Find $TU$

Now that we know $n = 3$, substitute this value back into the expression for $TU$:

$TU = 5n + 7 = 5(3) + 7 = 15 + 7 = 22$

Final Answer:

$TU = 22$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the problem change if point $T$ were not between $S$ and $U$?
2. Can you find the lengths of $SU$ and $ST$ when $n = 3$?
3. What if $n$ were negative? Would the segment lengths still make sense?
4. How do you verify if the solution is consistent with the condition $T$ is between $S$ and $U$?
5. What happens to the segment lengths if you double the value of $n$?

Tip: When solving problems involving points on a line, always remember to set up an equation that reflects the sum of the segments if the points are collinear.