If 3,A,B,192 are consecutive term of a GP,find the value of A&B.
Well, in order to find the value of A and B, let's look at the sequence 3, A, B, 192.
Now, since these terms are in a geometric progression (GP), we know that there is a common ratio between consecutive terms. Let's call that common ratio "r".
So, based on that, we have:
A = 3 * r
B = A * r = 3 * r * r
Now, we also know that B = 192. Let's substitute that in:
3 * r * r = 192
Solving this equation, we find that r = 8.
Now we can substitute this value of r into the equations for A and B:
A = 3 * r = 3 * 8 = 24
B = A * r = 24 * 8 = 192
So, the value of A is 24, and the value of B is also 192.
Hope that puts a smile on your face!
To find the value of A and B in the geometric progression (GP), we can use the formula for finding the nth term of a GP.
The formula for the nth term in a GP is:
An = A * r^(n-1)
where An is the nth term, A is the first term, r is the common ratio, and n is the position of the term.
Given that 3, A, B, and 192 are consecutive terms, we can determine their positions in the sequence:
3 is the first term (A) with n = 1,
A is the second term with n = 2,
B is the third term with n = 3,
192 is the fourth term with n = 4.
Using the formula, we can set up two equations:
3A = A * r^(2-1) [equation 1]
AB = A * r^(3-1) [equation 2]
Dividing equation 2 by equation 1:
(AB)/(3A) = (A * r^(3-1))/(A * r^(2-1))
B/3 = r
B = 3r [equation 3]
Substituting equation 3 into equation 2:
A * 3r = A * r^(3-1)
3r = r^2
r^2 - 3r = 0
r(r - 3) = 0
From this equation, we have two possible solutions for r:
r = 0 or r = 3.
If r = 0, the terms of the GP would all be equal to zero, which is not possible since 3 and 192 are given terms.
If r = 3, we can substitute this value back into equation 3:
B = 3r
B = 3 * 3
B = 9
Therefore, the value of A is 3 and the value of B is 9.
To find the value of A and B, we first need to understand the concept of a geometric progression (GP). In a GP, each term is obtained by multiplying the previous term by a constant factor called the common ratio (r).
Given that 3, A, B, and 192 are consecutive terms of a GP, we can write the following equations:
A = 3 * r
B = A * r
192 = B * r
Let's solve these equations step by step to find the values of A and B.
1. Substitute the value of A in terms of r into the equation A = 3 * r:
3 * r = 3 * r
This equation shows that the value of A in terms of r is satisfied.
2. Substitute the value of A in terms of r into the equation B = A * r:
B = (3 * r) * r
B = 3 * r^2
3. Substitute the value of B in terms of r into the equation 192 = B * r:
192 = (3 * r^2) * r
192 = 3 * r^3
Now, we have a cubic equation. Let's solve it to find the value of r.
Divide both sides of the equation by 3:
64 = r^3
Take the cube root of both sides:
∛64 = ∛(r^3)
4 = r
We have found the value of r, which is 4.
Now, substitute this value of r into the equations for A and B:
A = 3 * r
A = 3 * 4
A = 12
B = A * r
B = 12 * 4
B = 48
Therefore, the value of A is 12 and the value of B is 48.
3,A,B,192 are a GP
a = 3 and by definition:
ar^3 = 192
3r^3 = 192
r^3 = 64
r = 4
your GP is 3, 12, 48, 192