Variables and Algebraic Expressions: Regularity

To identify which expressions represent a term-to-term rule for a sequence, we'll simplify each expression:

• Expression a: $5n + 1 - 3n$
  Simplifying this, we combine like terms:
  $(5n - 3n) + 1 = 2n + 1$.
  This simplifies to a linear expression: $2n + 1$.
• Expression b: $2n + 1$
  This expression is already in its simplest linear form.
• Expression c: $7n - 1 - 5n$
  Simplifying this, we combine like terms:
  $(7n - 5n) - 1 = 2n - 1$.
  This simplifies to a linear expression: $2n - 1$.
• Expression d: $1 - 4n + 6n$
  Simplifying this, we combine like terms:
  $1 + (6n - 4n) = 1 + 2n$.
  This also simplifies to a linear expression: $2n + 1$.

After simplification:

• Expression a simplifies to $2n + 1$.
• Expression b is already $2n + 1$.
• Expression c simplifies to $2n - 1$, which is not identical to $2n + 1$ and does not match the sequence rule form identified in expressions a, b, and d.
• Expression d simplifies to $2n + 1$.

Thus, the expressions that represent a term-to-term rule of the form $2n + 1$ are

a, b, and d

To solve this problem, we'll evaluate each expression:

• Expression a: $9n^2 + 5n - 8n^2 + 2 - 6n$
  Simplify by combining like terms: $(9n^2 - 8n^2) + (5n - 6n) + 2 = n^2 - n + 2$. This simplification gives a leading positive quadratic term.

• Expression b: $3n + 2$
  It is a linear polynomial, which represents a constant positive growth. Thus, it could model growth, but we will further compare it with others to establish viable growth forms relevant to plant growth requirements.

• Expression c: $n^2 - n + 2$
  Already simplified; exhibits growth as $n$ increases due to the positive leading coefficient of $n^2$.

• Expression d: $1 + n^2 + 2n^2 - 3n^2 + 1$
  Simplify: $(n^2 + 2n^2 - 3n^2) + (1 + 1) = 2$. This simplifies to a constant, not representing increasing growth.

• Expression e: $-5 + 3n^2 - 7n + 4$
  Simplify: $3n^2 - 7n - 1$. Although quadratic and can represent growth, the constant term seems irrelevant for shrub growth understood here.

• Expression f: $9n^2 - 3n - 4$. Already in standard form, offering similar growth properties to “c” in pure quadratic format but compared to a and c it edges. Yet, bears distractions in constants.

The expressions suitable for showing flower growth are those with positive quadratic terms and constant growth interpreted about factors involved in the task relating floral growth to equations suitable. Therefore, expression a and c illustrate this better by showing proper polynomial growth visualized for floristic relevance. Thus, the answer is a, c.

To find the expression that accurately represents the sequence, let's analyze the given numbers:

• The first term is $2$, which can be represented as $2 + \frac{1}{1} = 2 + \frac{1}{1}$ for $n = 2$.
• The second term is $2\frac{1}{2} = 2 + \frac{1}{2}$, when $n = 2$.
• The third term is $3\frac{1}{3} = 3 + \frac{1}{3}$, when $n = 3$.
• The fourth term is $4\frac{1}{4} = 4 + \frac{1}{4}$, when $n = 4$.

From the sequence pattern, we see that each term is indeed $n + \frac{1}{n}$.

Now, let's express each choice based on $n$:

• Choice 1: $2n + \frac{1}{n}$ - This does not match our pattern.
• Choice 2: $\frac{1}{0.5n} + 2n + \frac{1}{0.5n} - n$ - This form is complex and incorrect.
• Choice 3: Indicates there is no correct property, but we identified one.
• Choice 4: $n^2+\frac{2}{n}+3n-n^2-\frac{1}{n}-2n$ simplifies to $n + \frac{1}{n}$.

Therefore, the expression $n^2+\frac{2}{n}+3n-n^2-\frac{1}{n}-2n$ simplifies correctly to describe the term-to-term rule of the sequence.

The solution to the problem is the expression: $n^2+\frac{2}{n}+3n-n^2-\frac{1}{n}-2n$.

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.