If 5th and 8th terms of A GP are 80 and 640, what's the common ratio?
T8/T5 = r^3 = 8
r = 2
To find the common ratio in a geometric progression (GP), we can use the formula:
\[a_n = a_1 \times r^{(n-1)}\]
where:
\[a_n\] is the nth term of the GP
\[a_1\] is the first term of the GP
\[r\] is the common ratio of the GP
Given that the 5th term of the GP is 80 and the 8th term is 640, we can set up two equations using the formula and the given values.
For the 5th term:
\[80 = a_1 \times r^{(5-1)}\]
Simplifying this equation, we get:
\[80 = a_1 \times r^4\] --- Equation 1
For the 8th term:
\[640 = a_1 \times r^{(8-1)}\]
Simplifying this equation, we get:
\[640 = a_1 \times r^7\] --- Equation 2
Now, we need to solve these two equations simultaneously to find the values of \(a_1\) and \(r\).
Dividing Equation 2 by Equation 1, we can eliminate \(a_1\) and solve for \(r\):
\[\frac{640}{80} = \frac{a_1 \times r^7}{a_1 \times r^4}\]
Simplifying, we get:
\[8 = r^3\]
Taking the cube root of both sides, we find:
\[r = 2\]
Therefore, the common ratio of the geometric progression is 2.