Find the value of k if 3,k, 48 are in g.p.
Again, use your definition:
k/3 = 48/k
k^2 = 144
k = ± 12
A=3
r=k÷3
An=arn-1=
=3×k÷3n-1
=Kn-1
Ah, finding the value of k in a geometric progression, how exciting! Let's put on our mathematical clown noses and get to work!
In a geometric progression, each term is obtained by multiplying the previous term by a common ratio. So if we have 3, k, and 48 in a geometric progression, we can write it as:
3, 3 * r, 3 * r^2 = 3, k, 48
Now, to find the common ratio (r), we can divide any term by the previous term. Let's divide k by 3:
(k / 3) = (3 * r^2) / (3 * r)
Simplifying this equation a little, we get:
k / 3 = r
So, the common ratio (r) is equal to k divided by 3. We can substitute this back into the original equation:
3 * r^2 = 48
Substituting r with k/3, we get:
3 * (k/3)^2 = 48
Simplifying further:
(k^2) / 3 = 16
Multiplying both sides by 3:
k^2 = 48
Now, take the square root of both sides:
k = √48
Time for a drumroll, please! *drumroll*
The value of k, in this case, is the square root of 48!
So, k ≈ 6.93.
And there you have it! The value of k in the geometric progression 3, k, 48 is approximately 6.93. I hope this answer tickled your funny bone as much as it helped you find the value of k!
In a geometric progression (G.P.), each term is obtained by multiplying the previous term by a common ratio.
Let's assume that the first term of the G.P. is a, and the common ratio is r.
According to the given information, the terms of the G.P. are 3, k, and 48.
Using the formula for the terms of a G.P., we can write the following equations:
k = 3 * r ----- (Equation 1)
48 = k * r ----- (Equation 2)
To find the value of k, we need to solve these equations simultaneously.
By substituting Equation 1 into Equation 2, we get:
48 = (3 * r) * r
48 = 3r^2
Dividing both sides of the equation by 3 gives:
16 = r^2
Taking the square root of both sides gives:
r = ±4
Now, substitute the value of r back into Equation 1 to solve for k:
k = 3 * r
If r = 4:
k = 3 * 4
k = 12
If r = -4:
k = 3 * -4
k = -12
So, the values of k can be either 12 or -12, depending on the value of the common ratio (r).
To find the value of k in the given geometric progression (g.p.), we need to use the formula for the nth term of a g.p., which is:
an = a1 * r^(n-1)
where:
- an is the nth term,
- a1 is the first term, and
- r is the common ratio between consecutive terms.
In this case, we know the first term (3) and the third term (48). We are trying to find the second term (k) and the common ratio. Let's substitute the values we know into the formula:
48 = 3 * r^(3-1)
To simplify this equation further, we can write 3 as 3 * r^(2). Now our equation becomes:
48 = 3 * r^2
To solve for r, we divide both sides of the equation by 3:
r^2 = 48 / 3
r^2 = 16
Taking the square root of both sides gives us:
r = ± √16
r = ± 4
Since a geometric progression cannot have a negative common ratio, we use the positive value:
r = 4
Now, to find k, we substitute the value of r into the equation for the nth term:
k = a1 * r^(n-1)
In this case, n is 2 (the second term), a1 is 3 (the first term), and r is 4:
k = 3 * 4^(2-1)
k = 3 * 4^1
k = 3 * 4
k = 12
Therefore, the value of k is 12 in the given geometric progression.