The third term of a G. P is 9 and the fifth term is 16. Find the fourth term of an G. P.
16 / f = f / 9
To find the fourth term of a geometric progression (G.P.), we need to determine the common ratio (r).
Let's denote the first term of the G.P. as 'a', and the common ratio as 'r'.
Given that the third term of the G.P. is 9, we can write:
a * r^2 = 9 ...(1)
Given that the fifth term of the G.P. is 16, we can write:
a * r^4 = 16 ...(2)
To eliminate 'a' from the above equations, divide equation (2) by equation (1):
(r^4) / (r^2) = 16 / 9
r^2 = (16 / 9)
r^2 = 1.7778
Taking the square root of both sides, we find:
r = √(1.7778)
r ≈ 1.3333
Now let's substitute the value of 'r' into equation (1) to find the value of 'a':
a * (1.3333)^2 = 9
1.7778a = 9
a ≈ 9 / 1.7778
a ≈ 5.0625
Therefore, the first term (a) of the G.P. is approximately 5.0625 and the common ratio (r) is approximately 1.3333.
To find the fourth term, we can use the formula:
Fourth term = a * r^3
Substituting the values, we get:
Fourth term ≈ 5.0625 * (1.3333)^3
Fourth term ≈ 5.0625 * 2.3703
Fourth term ≈ 11.965
Hence, the fourth term of the G.P. is approximately 11.965.