The third term of a GP is 9 and the fifth term is 16. Find the fourth term.
To find the fourth term of the geometric progression (GP), we need to first find the common ratio of the sequence.
In a geometric progression, each term is obtained by multiplying the previous term by a constant factor called the common ratio (r).
Let's denote the first term as 'a', the second term as 'ar', the third term as 'ar^2', and so on.
Given that the third term is 9, we have:
ar^2 = 9 ----- (1)
Also, given that the fifth term is 16, we have:
ar^4 = 16 ----- (2)
Dividing equation (2) by equation (1) will help us find the value of the common ratio (r):
(ar^4) / (ar^2) = 16 / 9
Cancelling out the common term 'ar^2', we get:
r^2 = (16 / 9)
Taking the square root of both sides, we have:
r = sqrt(16 / 9)
Simplifying the square root:
r = 4 / 3
Now that we have the value of the common ratio (r), we can find the fourth term by multiplying the third term by the common ratio:
fourth term = ar^3
Substituting the known values:
fourth term = 9 * (4 / 3)^3
Simplifying the expression:
fourth term = 9 * (64 / 27)
Multiplying the numerator and denominator:
fourth term = (9 * 64) / 27
Calculating the value:
fourth term = 576 / 27
Simplifying the fraction:
fourth term = 64 / 3
Therefore, the fourth term of the geometric progression is 64/3 or approximately 21.333.
a * r^2 = 9
a * r^4 = 16
dividing ... r^2 = 16 / 9 ... r = 4 / 3
9 * 4/3 = ?