1; 3; 9; 27;... Which statement best describes the pattern?

21. Step 1: Examine the pattern given: $1; 3; 9; 27; ...$ Step 2: Check the relationship between consecutive terms. Divide the second term by the first: $\frac{3}{1} = 3$. Divide the third term by the second: $\frac{9}{3} = 3$. Divide the fourth term by the third: $\frac{27}{9} = 3$. Step 3: Since there is a constant ratio of $3$ between consecutive terms, the statement that best describes the pattern is that a constant ratio of three is used.

The correct option is A.

22. Step 1: Analyze the given table to find the relationship between $n$ and $T_n$. n | 1 | 3 | 5 | 7 Tn | 5 | 1 | -3 | -7 Step 2: Calculate the change in $T_n$ for a given change in $n$. When $n$ changes from $1$ to $3$ (change of $+2$), $T_n$ changes from $5$ to $1$ (change of $-4$). When $n$ changes from $3$ to $5$ (change of $+2$), $T_n$ changes from $1$ to $-3$ (change of $-4$). This indicates a linear relationship of the form $T_n = mn + c$. The slope $m = \frac{change in T_n}{change in n} = \frac{-4}{2} = -2$. Step 3: Substitute the slope $m = -2$ into the equation $T_n = -2n + c$. Use one of the points from the table, for example, $(n=1, T_n=5)$. $5 = -2(1) + c$ $5 = -2 + c$ $c = 5 + 2$ $c = 7$ Step 4: Write the general term using the calculated $m$ and $c$. $T_n = -2n + 7$.

The correct option is C.

23. Step 1: Identify the given expression: $3x^3 + 3x(x^2) + 3x^2 + 3x$. Step 2: Simplify any terms in the expression. The term $3x(x^2)$ can be simplified to $3x^{1+2} = 3x^3$. So the expression becomes $3x^3 + 3x^3 + 3x^2 + 3x$. Step 3: Recall that like terms have the same variable(s) raised to the same power(s). From the simplified expression, the terms are $3x^3$, $3x^3$, $3x^2$, and $3x$. Step 4: Check the given options for pairs of like terms. A $3x^3$ and $3x(x^2)$: Since $3x(x^2) = 3x^3$, this pair consists of $3x^3$ and $3x^3$, which are like terms.

The correct option is A.

24. Step 1: Identify the given term: $(-2x^2)^3$. Step 2: Apply the exponent to both the coefficient and the variable part. $(-2x^2)^3 = (-2)^3 \cdot (x^2)^3$ Step 3: Calculate the numerical part and the variable part separately. $(-2)^3 = -2 \times -2 \times -2 = -8$. $(x^2)^3 = x^{2 \times 3} = x^6$. Step 4: Combine the results to get the simplified term. The simplified term is $-8x^6$. Step 5: Identify the coefficient and the degree of the term. The coefficient is the numerical factor, which is $-8$. The degree of the term is the exponent of the variable, which is $6$.

The correct option is D.

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