The third and fifth terms of a geometric progression are 9/2 and 81/8 respectively, find the common ratio and first term
Let's denote the common ratio of the geometric progression as 'r' and the first term as 'a'.
We know that the third term is 9/2, so we can express it as:
a * r^2 = 9/2
Similarly, the fifth term is 81/8, so we can express it as:
a * r^4 = 81/8
To solve this system of equations, let's first rewrite the fractions with a common denominator:
a * r^2 = 36/8
a * r^4 = 81/8
Next, let's simplify the equations:
a * r^2 = 9/2
a * r^4 = 9
Now, let's divide the second equation by the first equation to eliminate 'a':
(a * r^4) / (a * r^2) = 9 / (9/2)
r^2 = 2
Taking the square root of both sides, we find:
r = √2
Now, let's substitute the value of 'r' into one of the original equations to solve for 'a'. Using the third equation:
a * r^2 = 9/2
a * (√2)^2 = 9/2
a * 2 = 9/2
a = (9/2) / 2
a = 9/4
Therefore, the common ratio is √2 and the first term is 9/4.