The 6th Term Of A Gp Is-2/27 And Its First Term Is 18.What Is The Common Ratio
18 r^5 = 2/27
9 r^5 = 1/27
r^5 = 3^-3 / 3^2 = 3^-5
math
To find the common ratio in a geometric progression (GP), we can use the formula:
an = a1 * r^(n-1)
where 'an' represents the nth term, 'a1' is the first term, 'r' is the common ratio, and 'n' is the term number.
Given that the 6th term (a6) is -2/27 and the first term (a1) is 18, we can substitute these values into the formula and solve for the common ratio (r):
-2/27 = 18 * r^(6-1)
Simplifying the equation, we have:
-2/27 = 18 * r^5
Divide both sides of the equation by 18 to isolate the term with the variable:
(-2/27) / 18 = r^5
Simplifying further:
-2/486 = r^5
Taking the fifth root of both sides, we can find the value of 'r':
r = (−2/486)^(1/5)
Calculating this value, we get:
r ≈ -0.3333
Therefore, the common ratio in this geometric progression is approximately -0.3333.
To find the common ratio of a geometric progression (GP), we can use the formula:
\[a_n = a_1 \cdot r^{(n-1)}\]
where \(a_n\) represents the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
Given that the first term (\(a_1\)) is 18 and the 6th term (\(a_6\)) is -2/27, we can substitute these values into the formula to find the common ratio.
\[a_6 = 18 \cdot r^{(6-1)}\]
Simplifying further:
\[-\frac{2}{27} = 18 \cdot r^{5}\]
Now, we can isolate the common ratio (r) by dividing both sides of the equation by 18:
\[\frac{-2}{27 \cdot 18} = r^{5}\]
Simplifying the left side:
\[\frac{-1}{243} = r^{5}\]
To find the common ratio (r), we will take the fifth root of both sides:
\[r = \left(\frac{-1}{243}\right)^{\frac{1}{5}}\]
Calculating the right side:
\[r = \left(\frac{-1}{3^5}\right)^{\frac{1}{5}}\]
Simplifying further:
\[r = \frac{1}{\sqrt[5]{3^5}}\]
Finally:
\[r = \frac{1}{3}\]
Therefore, the common ratio (r) of the geometric progression is 1/3.