Find the 12th term of the geometric sequence shown below. 2x^(2),4x^(4),8x^(6),dots Answer:

Identify Pattern: Identify the pattern in the sequence. The given sequence is 2x^2, 4x^4, 8x^6, ... We can see that each term is obtained by multiplying the previous term by 2x^2.Determine Common Ratio: Determine the common ratio (r) of the sequence. Since each term is obtained by multiplying the previous term by 2x^2, the common ratio r is 2x^2.Use Formula for nth Term: Use the formula for the nth term of a geometric sequence. The nth term (a_n) of a geometric sequence can be found using the formula a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.Substitute Values: Substitute the values into the formula to find the 12th term. The first term a_1 is 2x^2 and the common ratio r is 2x^2. We want to find the 12th term, so n = 12. a_12 = 2x^2 * (2x^2)^(12-1)Simplify Expression: Simplify the expression for the 12th term. a_12 = 2x^2 * (2x^2)^11 a_12 = 2x^2 * 2^11 * x^(2*11) a_12 = 2x^2 * 2048 * x^22 a_12 = 4096 * x^(2+22) a_12 = 4096 * x^24

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Q. Find the

12

th term of the geometric sequence shown below.

\newline

2 x^{2}, 4 x^{4}, 8 x^{6}, \ldots

\newline

Answer:

Identify Pattern: Identify the pattern in the sequence. $\newline$ The given sequence is $2x^2$ , $4x^4$ , $8x^6$ , ... $\newline$ We can see that each term is obtained by multiplying the previous term by $2x^2$ .
Determine Common Ratio: Determine the common ratio $r$ of the sequence. $\newline$ Since each term is obtained by multiplying the previous term by $2x^2$ , the common ratio $r$ is $2x^2$ .
Use Formula for nth Term: Use the formula for the nth term of a geometric sequence. $\newline$ The nth term $a_n$ of a geometric sequence can be found using the formula $a_n = a_1 \cdot r^{(n-1)}$ , where $a_1$ is the first term and $r$ is the common ratio.
Substitute Values: Substitute the values into the formula to find the $12^{\text{th}}$ term. $\newline$ The first term $a_1$ is $2x^2$ and the common ratio $r$ is $2x^2$ . We want to find the $12^{\text{th}}$ term, so $n = 12$ . $\newline$ $a_{12} = 2x^2 \cdot (2x^2)^{12-1}$
Simplify Expression: Simplify the expression for the $12$ th term. $\newline$ $a_{12} = 2x^2 \times (2x^2)^{11}$ $\newline$ $a_{12} = 2x^2 \times 2^{11} \times x^{2\times11}$ $\newline$ $a_{12} = 2x^2 \times 2048 \times x^{22}$ $\newline$ $a_{12} = 4096 \times x^{2+22}$ $\newline$ $a_{12} = 4096 \times x^{24}$