The third term of a GP is 9 and the fifth term is 16. find the 4th term.
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To find the fourth term of a geometric progression (GP), we need to understand the relationship between the terms.
A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant value called the common ratio (r).
Let's denote the first term as 'a', the common ratio as 'r', and the fourth term as 'T4'.
We are given the following information:
The third term (T3) is 9, which means a * r^2 = 9.
The fifth term (T5) is 16, which means a * r^4 = 16.
Now, we can solve these two equations to find the values of 'a' and 'r' by dividing the second equation by the first equation:
(a * r^4) / (a * r^2) = 16 / 9
Simplifying the equation, we get:
r^2 = (16 / 9)
Taking the square root of both sides, we find:
r = sqrt(16 / 9)
r = (4 / 3)
Now, substituting the value of 'r' into the equation a * r^2 = 9, we can solve for 'a':
a * (4/3)^2 = 9
a * (16/9) = 9
Multiplying both sides by (9/16), we get:
a = (9/16) * (16/9)
a = 1
So, the first term (a) is 1, and the common ratio (r) is 4/3.
Finally, we can find the fourth term (T4) by multiplying the first term 'a' by the common ratio 'r' raised to the power of 3 (since the fourth term follows the pattern a * r^3):
T4 = a * r^3
T4 = 1 * (4/3)^3
Calculating this, we find:
T4 = 1 * (64/27) = 64/27
Therefore, the fourth term of the geometric progression is 64/27.