Find the 12th term of the geometric sequence shown below. 6x^(6),-30x^(11),150x^(16),dots Answer:

Find Common Ratio: To find the 12th term of a geometric sequence, we need to identify the common ratio (r) between consecutive terms. We can find the common ratio by dividing the second term by the first term. Calculation: r = (-30x^(11)) / (6x^(6)) r = -5x^(11-6) r = -5x^5Calculate 12th Term: Now that we have the common ratio, we can find the 12th term (a12) using the formula for the nth term of a geometric sequence, which is an = a1 * r^(n-1), where a1 is the first term and n is the term number. Calculation: a12 = 6x^(6) * (-5x^5)^(12-1) a12 = 6x^(6) * (-5x^5)^(11)Simplify Expression: We need to simplify the expression for a12 by performing the exponentiation and multiplication. Calculation: a12 = 6x^(6) * (-5)^11 * x^(5*11) a12 = 6x^(6) * (-5)^11 * x^(55) a12 = 6 * (-5)^11 * x^(6+55) a12 = 6 * (-5)^11 * x^(61)Calculate Value: Finally, we calculate the value of (-5)^11 to find the 12th term. Calculation: (-5)^11 = -5 * -5 * -5 * -5 * -5 * -5 * -5 * -5 * -5 * -5 * -5 (-5)^11 = -48828125 a12 = 6 * (-48828125) * x^(61) a12 = -292968750 * x^(61)

Home

Math Problems

Algebra 1

Evaluate variable expressions for number sequences

Full solution

Q. Find the

12

th term of the geometric sequence shown below.

\newline

6 x^{6},-30 x^{11}, 150 x^{16}, \ldots

\newline

Answer:

Find Common Ratio: To find the $12^{\text{th}}$ term of a geometric sequence, we need to identify the common ratio ( $r$ ) between consecutive terms. We can find the common ratio by dividing the second term by the first term. $\newline$ Calculation: $\newline$ $r = \frac{-30x^{11}}{6x^{6}}$ $\newline$ $r = -5x^{11-6}$ $\newline$ $r = -5x^5$
Calculate $12$ th Term: Now that we have the common ratio, we can find the $12$ th term $a_{12}$ using 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: $\newline$ $a_{12} = 6x^{6} \cdot (-5x^5)^{12-1}$ $\newline$ $a_{12} = 6x^{6} \cdot (-5x^5)^{11}$
Simplify Expression: We need to simplify the expression for $a_{12}$ by performing the exponentiation and multiplication. $\newline$ Calculation: $\newline$ $a_{12} = 6x^{6} \times (-5)^{11} \times x^{5\times11}$ $\newline$ $a_{12} = 6x^{6} \times (-5)^{11} \times x^{55}$ $\newline$ $a_{12} = 6 \times (-5)^{11} \times x^{6+55}$ $\newline$ $a_{12} = 6 \times (-5)^{11} \times x^{61}$
Calculate Value: Finally, we calculate the value of $(-5)^{11}$ to find the $12$ th term. $\newline$ Calculation: $\newline$ $(-5)^{11} = -5 \times -5 \times -5 \times -5 \times -5 \times -5 \times -5 \times -5 \times -5 \times -5 \times -5$ $\newline$ $(-5)^{11} = -48828125$ $\newline$ $a_{12} = 6 \times (-48828125) \times x^{61}$ $\newline$ $a_{12} = -292968750 \times x^{61}$