For the geometric sequence t: {3, m, n, 192, . . .}, find the values for m and n.(Answer is m=12, n=48)
How can we do this without trial and error? Thanks to anyone who helps.
You have the first and the fourth term of the sequence. If you make equations for the first and fourth term you can use them to solve for r, and then sub it back into the original to find the second and third terms : )
You have the first and the fourth term of the sequence. If you make equations for the first and fourth term you can use them to solve for r, and then sub it back into the original to find the second and third terms : )
like this:
m/3 = n/m
m^2 =3n ---> m^4 = 9n^2
n/m = 192/n
n^2 = 192m
using m^4 = 9n^2
m^4 = 9(192m)
m^4 - 1728m = 0
m(m^3 - 1728) = 0
m = 0 , which it can't, since we would be dividing by 0
or
m^3 = 1728 , m = 12
in n^2 = 192m
n^2 = 192(12) = 2304
n = 48
To find the values of m and n in the geometric sequence {3, m, n, 192, ...}, we need to use the properties of a geometric sequence and its common ratio.
A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio (r). The general form of a geometric sequence is {a, ar, ar^2, ar^3, ...}, where a is the first term and r is the common ratio.
In this case, we know that the sequence starts with 3, so the first term (a) is 3. We need to determine the common ratio (r) and subsequently find the values of m and n.
To find the common ratio, we can divide any term in the sequence by the previous term. Let's divide the third term n by the second term m.
n / m = (ar^2) / (ar) = r
Since the common ratio (r) is the same for all terms in a geometric sequence, we can also divide the fourth term (192) by the third term (n) to find the value of r.
192 / n = (ar^3) / (ar^2) = r
So, we have two equations:
1) n / m = r
2) 192 / n = r
To avoid trial and error, we can set these two equations equal to each other and solve for n in terms of m:
n / m = 192 / n
By cross-multiplying, we get:
n^2 = 192m
Now, let's substitute the value of n^2 from equation 2 into equation 1:
192m = r^2m
Since the r value is the same in both equations, we can eliminate it by dividing both sides by m:
192 = r^2
To find the value of r, we can take the square root of both sides:
sqrt(192) = r
Simplifying the square root of 192, we get:
r ≈ 13.856
Now that we know the value of r, we can find the value of m:
n / m = r
n = rm
Substituting the value of r and rearranging the equation, we have:
48 = 13.856m
Finally, to solve for m, we divide both sides by 13.856:
m ≈ 3.453
Therefore, the approximate values for m and n are m = 3.453 and n = 48.