Find the 12th term of the geometric sequence shown below. x^(3),3x^(4),9x^(5),dots Answer:

Calculate Common Ratio: Identify the common ratio (r) of the geometric sequence by dividing the second term by the first term. Calculation: r = (3x^(4)) / (x^(3)) = 3x^(4-3) = 3xVerify Consistency: Verify that the common ratio remains consistent by dividing the third term by the second term. Calculation: r = (9x^(5)) / (3x^(4)) = 3x^(5-4) = 3xUse nth Term Formula: Use the formula for the nth term of a geometric sequence, which is a_n = a_1 * r^(n-1), where a_1 is the first term and n is the term number. Calculation: a_12 = x^(3) * (3x)^(12-1)Simplify 12th Term: Simplify the expression for the 12th term by calculating the exponent. Calculation: a_12 = x^(3) * (3x)^(11) = x^(3) * 3^(11) * x^(11) = 3^(11) * x^(3+11) = 3^(11) * x^(14)

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Q. Find the

12

th term of the geometric sequence shown below.

\newline

x^{3}, 3 x^{4}, 9 x^{5}, \ldots

\newline

Answer:

Calculate Common Ratio: Identify the common ratio $r$ of the geometric sequence by dividing the second term by the first term. $\newline$ Calculation: $r = \frac{3x^{4}}{x^{3}} = 3x^{4-3} = 3x$
Verify Consistency: Verify that the common ratio remains consistent by dividing the third term by the second term. $\newline$ Calculation: $r = \frac{9x^{5}}{3x^{4}} = 3x^{5-4} = 3x$
Use nth Term Formula: Use the formula for the nth term of a geometric sequence, which is $a_n = a_1 \cdot r^{(n-1)}$ , where $a_1$ is the first term and $n$ is the term number. $\newline$ Calculation: $a_{12} = x^{3} \cdot (3x)^{12-1}$
Simplify $12$ th Term: Simplify the expression for the $12$ th term by calculating the exponent. $\newline$ Calculation: $a_{12} = x^{3} \times (3x)^{11} = x^{3} \times 3^{11} \times x^{11} = 3^{11} \times x^{3+11} = 3^{11} \times x^{14}$