b) y=±1416(x)Find the common ratio of the geometric sequence {an}n=1∞ given that:\begin{align*}\(\newline&\left\{\begin{array}{l}-\frac{1}{4}=a_{1}+d \quad d=-\frac{1}{4}-a_{1},(\newline\)a_{2}=-\frac{1}{4},(\newline\)a_{6}=-\frac{12}{243}\end{array}\right.\end{align*}\)
Q. b) y=±1416(x)Find the common ratio of the geometric sequence {an}n=1∞ given that:\begin{align*}\(\newline&\left\{\begin{array}{l}-\frac{1}{4}=a_{1}+d \quad d=-\frac{1}{4}-a_{1},(\newline\)a_{2}=-\frac{1}{4},(\newline\)a_{6}=-\frac{12}{243}\end{array}\right.\end{align*}\)
Identify terms: Identify the first term a1 and the sixth term a6 of the geometric sequence.Given: a1=−41 and a6=−24312.
Use formula for nth term: Recall the formula for the nth term of a geometric sequence: an=a1×r(n−1), where r is the common ratio.We need to find r using the given terms.
Express a6 in terms: Use the formula to express a6 in terms of a1 and r. a6=a1⋅r6−1=a1⋅r5
Substitute given values: Substitute the given values for a1 and a6 into the equation.−24312=(−41)⋅r5
Solve for r5: Solve for r5 by dividing both sides of the equation by a1.r5=−1/4−12/243
Calculate r5: Calculate r5.r5=243−12×1−4=24348
Simplify fraction: Simplify the fraction 24348. r5=24348=8116
Find common ratio r: Find the fifth root of 8116 to get the common ratio r.r=(8116)51
Calculate fifth root: Calculate the fifth root of 8116. r=(32)51