We are asked to find the 20th term of an arithmetic sequence where:

• The 3rd term is $a_3 = 7$,
• The 7th term is 2 more than three times the 3rd term: $a_7 = 3 \times a_3 + 2$.

Step 1: Express terms in terms of the first term $a_1$ and the common difference $d$.

For any arithmetic sequence, the $n$-th term is given by: $a_n = a_1 + (n - 1)d$

Thus, we can express the 3rd term and the 7th term in terms of $a_1$ and $d$:

• For the 3rd term: $a_3 = a_1 + 2d = 7$

• For the 7th term: $a_7 = a_1 + 6d$ But we are given that: $a_7 = 3 \times a_3 + 2 = 3 \times 7 + 2 = 21 + 2 = 23$

Step 2: Solve the system of equations.

Now we have two equations:

1. $a_1 + 2d = 7$
2. $a_1 + 6d = 23$

Subtract the first equation from the second:

$(a_1 + 6d) - (a_1 + 2d) = 23 - 7$ $4d = 16$ $d = 4$

Substitute $d = 4$ into the first equation:

$a_1 + 2(4) = 7$ $a_1 + 8 = 7$ $a_1 = -1$

Step 3: Find the 20th term.

Now that we know $a_1 = -1$ and $d = 4$, we can find the 20th term using the formula for the $n$-th term:

$a_{20} = a_1 + (20 - 1)d = -1 + 19 \times 4 = -1 + 76 = 75$

Thus, the 20th term is $\boxed{75}$.

Would you like further details or have any questions?

Here are some related problems:

1. Find the 10th term of the same sequence.
2. Determine the sum of the first 20 terms.
3. What is the common difference if the 4th and 8th terms are given?
4. Find the first term if the 5th and 9th terms are known.
5. How does the common difference affect the growth of the sequence?

Tip: Always check if the terms of the sequence form a pattern by calculating the difference between consecutive terms.