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Find the 12th term of the geometric sequence shown below.

-8x^(2),16x^(5),-32x^(8),dots
Answer:

Find the $12$ th term of the geometric sequence shown below. $\newline$ $-8 x^{2}, 16 x^{5},-32 x^{8}, \ldots$ $\newline$ Answer:

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Find trigonometric functions using a calculator

Full solution

Q. Find the

12

th term of the geometric sequence shown below.

\newline

-8 x^{2}, 16 x^{5},-32 x^{8}, \ldots

\newline

Answer:

Identify Common Ratio: To find the $12^{th}$ term of a geometric sequence, we need to identify the common ratio ( $r$ ) of the sequence. The common ratio is found by dividing any term by the previous term.
Calculate Common Ratio: Let's find the common ratio by dividing the second term by the first term: $r = \frac{16x^5}{-8x^2} = -2x^3$
Use Formula for nth Term: Now that we have the common ratio, we can use the formula for the nth term of a geometric sequence, which is $a_n = a_1 \times r^{(n-1)}$ , where $a_1$ is the first term and $n$ is the term number.
Find $12$ th Term: We want to find the $12$ th term $a_{12}$ . We already know the first term $a_1$ is $-8x^2$ and the common ratio $r$ is $-2x^3$ . Plugging these values into the formula gives us: $\newline$ $a_{12} = a_1 \cdot r^{12-1} = -8x^2 \cdot (-2x^3)^{11}$
Calculate Exponent: Now we need to calculate $(-2x^3)^{11}$ . When raising a power to a power, we multiply the exponents: $(-2x^3)^{11} = (-2)^{11} \times (x^3)^{11} = -2048x^{33}$
Multiply to Find $12$ th Term: Finally, we multiply the first term by this value to find the $12$ th term: $a_{12} = -8x^2 \times -2048x^{33} = 16384x^{35}$

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