Find the 8th term of the geometric sequence shown below. -9x^(6),-9x^(11),-9x^(16),dots Answer:

Identify Pattern: Identify the pattern in the sequence to determine the common ratio (r). The sequence is -9x^(6), -9x^(11), -9x^(16), ... where each term increases the exponent of x by 5. Calculate Common Ratio: Calculate the common ratio (r) by dividing the second term by the first term. r = (-9x^(11)) / (-9x^(6)) = x^(11-6) = x^5Use Geometric Sequence 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. Substitute Values for 8th Term: Substitute the values into the formula to find the 8th term. a_8 = (-9x^(6)) * (x^5)^(8-1)Simplify Exponent: Simplify the exponent in the formula. a_8 = (-9x^(6)) * (x^5)^7 a_8 = (-9x^(6)) * x^(5*7) a_8 = (-9x^(6)) * x^(35)Combine Exponents: Combine the exponents of x by adding them, since they have the same base. a_8 = -9 * x^(6+35) a_8 = -9 * x^(41)

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Q. Find the

8

th term of the geometric sequence shown below.

\newline

-9 x^{6},-9 x^{11},-9 x^{16}, \ldots

\newline

Answer:

Identify Pattern: Identify the pattern in the sequence to determine the common ratio $r$ . The sequence is $-9x^{6}$ , $-9x^{11}$ , $-9x^{16}$ , ... where each term increases the exponent of $x$ by $5$ .
Calculate Common Ratio: Calculate the common ratio $r$ by dividing the second term by the first term. $\newline$ $r = \frac{-9x^{11}}{-9x^{6}} = x^{11-6} = x^5$
Use Geometric Sequence Formula: Use the formula for the $n$ th 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.
Substitute Values for $8$ th Term: Substitute the values into the formula to find the $8$ th term. $\newline$ $a_8 = (-9x^{6}) \cdot (x^5)^{8-1}$
Simplify Exponent: Simplify the exponent in the formula. $\newline$ $a_8 = (-9x^{6}) \times (x^5)^7$ $\newline$ $a_8 = (-9x^{6}) \times x^{5\times7}$ $\newline$ $a_8 = (-9x^{6}) \times x^{35}$
Combine Exponents: Combine the exponents of $x$ by adding them, since they have the same base. $\newline$ $a_8 = -9 \times x^{(6+35)}$ $\newline$ $a_8 = -9 \times x^{41}$