Evaluate Term-to-Term Rules: Comparing 7n-2 vs 9n+4-2n-2 in Age Sequences

To solve this problem, we will evaluate each expression by substituting small values of $n$ (1, 2, and 3) and comparing the results to the given sequence 5, 12, 19, ...

We'll examine each option:

• Option a: $9n + 4 - 2n - 2$

The expression simplifies to $9n - 2n + 4 - 2 = 7n + 2$.

Substitute $n = 1,$ then $7(1) + 2 = 9.$ This does not match 5.

Hence, this rule is unsuitable.

• Option b: $x^2 + 5n - x^2 + 2n - 2$

This simplifies to $5n + 2n - 2 = 7n - 2$.

Substitute $n = 1,$ then $7(1) - 2 = 5,$ which matches.

Substitute $n = 2,$ then $7(2) - 2 = 12,$ which matches.

Substitute $n = 3,$ then $7(3) - 2 = 19,$ which matches.

This rule is suitable.

• Option c: $7n - 2$

This matches the rule used in option b.

Hence, this rule is suitable as well.

• Option d: $9n + 4 - n - 6 - n$

The expression simplifies to $9n - 2n + 4 - 6 = 7n - 2$.

This matches the results in options b and c when evaluated.

This rule is also suitable.

Therefore, the rules described in options b, c, and d generate the ages sequence correctly. All of these simplify to $7n - 2$.

The correct answer is choices b, d, and c.