The third term of a GP is 10 and the 6th term is 80.find the common ratio
r^(6 - 3) = 80 / 10 ... r^3 = 8
To find the common ratio (r) of a geometric progression (GP), we can use the formula:
\(a_n = a_1 \cdot r^{n-1}\)
Given that the third term (a₃) is 10 and the sixth term (a₆) is 80, we can set up the following equations:
\(10 = a_1 \cdot r^{3-1}\) --(1)
\(80 = a_1 \cdot r^{6-1}\) --(2)
Dividing equation (2) by equation (1), we can eliminate a₁:
\(\frac{80}{10} = \frac{a_1 \cdot r^{6-1}}{a_1 \cdot r^{3-1}}\)
This simplifies to:
8 = \(r^5\)
To solve for r, we can take the fifth root of both sides of the equation:
\(r = \sqrt[5]{8} = 2\)
Therefore, the common ratio (r) of the GP is 2.