If 3,p,q,24 are consecutive terms of an exponential sequence. Find the values of p and q.
Let the common ratio be r. Then we have:
3 * r = p
p * r = q
q * r = 24
Dividing the second equation by the first, we get:
p * r / (3 * r) = q / p
p / 3 = q / p
p^2 = 3q
Substituting this into the third equation, we get:
(3q) * r = 24
q * r = 8
Substituting q*r=8 into the second equation, we get:
p * r = 8
Dividing this equation by the first equation, we get:
(p * r) / (3 * r) = 8 / 3
p / 3 = 8 / 3
p = 8
Substituting p=8 into p^2=3q, we get:
8^2 = 3q
64 = 3q
q = 64/3
Therefore, the values of p and q are p = 8 and q = 64/3.
AAAaannndd the bot gets it wrong yet again!
r^3 = 24/3 = 8
so r = 2
The sequence is 3, 6, 12, 24
I'm sorry for the mistake in my previous response. You are correct that the common ratio is 2. Using this, we can find the values of p and q:
3 * 2 = 6, so p = 6
6 * 2 = 12, so q = 12
Therefore, the values of p and q in the sequence are p = 6 and q = 12.
To find the values of p and q in the exponential sequence, we need to find the common ratio (r) of the sequence.
The formula for finding the nth term in an exponential sequence is given by:
Tn = a * r^(n-1)
Where:
Tn = nth term
a = first term
r = common ratio
n = term number
In this case, the third term (T3) is given as 3, the fourth term (T4) is given as 24, and we need to find the values of p and q.
Step 1: Finding the common ratio
Using the equation Tn = a * r^(n-1), we can set up two equations based on the consecutive terms:
T3 = 3 = a * r^(3-1) = a * r^2
T4 = 24 = a * r^(4-1) = a * r^3
Dividing the second equation by the first equation, we get:
24/3 = (a * r^3) / (a * r^2)
8 = r
Step 2: Finding p and q
Since the common ratio (r) is 8, we can find p by substituting the known values into the equation:
3 = a * 8^2
3 = a * 64
a = 3/64
Now, we can find q by using the common ratio (r) and the value of a:
q = a * r
q = (3/64) * 8
q = 3/8
Therefore, the values of p and q in the exponential sequence are:
p = 3/64
q = 3/8
To solve this problem, we need to understand what an exponential sequence is and how it behaves.
An exponential sequence is a sequence in which each term is obtained by multiplying the previous term by a constant ratio. In other words, each term can be expressed as a = a1 * r^(n-1), where 'a' is the term, 'a1' is the first term, 'r' is the common ratio, and 'n' is the term number.
In this case, we are given that 3, p, q, and 24 are consecutive terms of an exponential sequence. Let's assign the first term, 3, to 'a1'. Since the terms are consecutive, we can assign 'n' as follows: 3 is the first term (n = 1), p is the second term (n = 2), q is the third term (n = 3), and 24 is the fourth term (n = 4).
So, for each term, we have:
a1 = 3
a2 = 3 * r^(2-1) = 3 * r
a3 = 3 * r^(3-1) = 3 * r^2
a4 = 3 * r^(4-1) = 3 * r^3
We also know that the consecutive terms are given as 3, p, q, and 24.
Therefore, we can write the equations as follows:
3 * r = p
3 * r^2 = q
3 * r^3 = 24
To find the values of p and q, we need to solve these equations.
From the first equation, we can express 'r' in terms of 'p': r = p / 3.
Substituting this in the second equation, we have: 3 * (p / 3)^2 = q
Simplifying, we get: p^2 = q
From the third equation, we have: 3 * (p / 3)^3 = 24
Simplifying, we get: p^3 = 24
Taking the square root of both sides of p^2 = q, we get: p = sqrt(q).
Substituting this value of p in the equation p^3 = 24, we have: (sqrt(q))^3 = 24
Simplifying, we get: sqrt(q^3) = 24
Taking the cube of both sides, we have: q^3 = 24^2 = 576
Taking the cube root of both sides, we get: q = 6
Substituting the value of q in the equation p = sqrt(q), we have: p = sqrt(6)
Therefore, the values of p and q are approximately p ≈ sqrt(6) and q ≈ 6.