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b) \newliney=±1614(x)y=\pm\frac{16}{14}(x)\newlineFind the common ratio of the geometric sequence \newline{an}n=1\{a_{n}\}_{n=1}^{\infty} given that:\newline\begin{align*}\(\newline&\left\{\begin{array}{l}\newline-\frac{1}{4}=a_{1}+d \quad d=-\frac{1}{4}-a_{1},(\newline\)a_{2}=-\frac{1}{4},(\newline\)a_{6}=-\frac{12}{243}\newline\end{array}\right.\newline\end{align*}\)

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Q. b) \newliney=±1614(x)y=\pm\frac{16}{14}(x)\newlineFind the common ratio of the geometric sequence \newline{an}n=1\{a_{n}\}_{n=1}^{\infty} given that:\newline\begin{align*}\(\newline&\left\{\begin{array}{l}\newline-\frac{1}{4}=a_{1}+d \quad d=-\frac{1}{4}-a_{1},(\newline\)a_{2}=-\frac{1}{4},(\newline\)a_{6}=-\frac{12}{243}\newline\end{array}\right.\newline\end{align*}\)
  1. Identify terms: Identify the first term a1a_1 and the sixth term a6a_6 of the geometric sequence.\newlineGiven: a1=14a_1 = -\frac{1}{4} and a6=12243a_6 = -\frac{12}{243}.
  2. Use formula for nth term: Recall the formula for the nth term of a geometric sequence: an=a1×r(n1)a_n = a_1 \times r^{(n-1)}, where rr is the common ratio.\newlineWe need to find rr using the given terms.
  3. Express a6a_6 in terms: Use the formula to express a6a_6 in terms of a1a_1 and rr.
    a6=a1r61=a1r5a_6 = a_1 \cdot r^{6-1} = a_1 \cdot r^5
  4. Substitute given values: Substitute the given values for a1a_1 and a6a_6 into the equation.\newline12243=(14)r5-\frac{12}{243} = \left(-\frac{1}{4}\right) \cdot r^5
  5. Solve for r5r^5: Solve for r5r^5 by dividing both sides of the equation by a1a_1.\newliner5=12/2431/4r^5 = \frac{-12/243}{-1/4}
  6. Calculate r5r^5: Calculate r5r^5.r5=12243×41=48243r^5 = \frac{-12}{243} \times \frac{-4}{1} = \frac{48}{243}
  7. Simplify fraction: Simplify the fraction 48243\frac{48}{243}. \newliner5=48243=1681r^5 = \frac{48}{243} = \frac{16}{81}
  8. Find common ratio rr: Find the fifth root of 1681\frac{16}{81} to get the common ratio rr.r=(1681)15r = \left(\frac{16}{81}\right)^{\frac{1}{5}}
  9. Calculate fifth root: Calculate the fifth root of 1681\frac{16}{81}. \newliner=(23)15r = \left(\frac{2}{3}\right)^{\frac{1}{5}}

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