If 3,p,q,24 are consecutive terms of an exponential sequence,find the value of p and q
Let the common ratio of the exponential sequence be r.
Then we know that:
3r = p
pr = q
qr = 24
We can use the first two equations to eliminate r:
3r * r = p * q
3r^2 = pq
And we can use the third equation to get an expression for r:
r = 24/q
Substituting this into the equation above, we get:
3(24/q)^2 = pq
1728/q^2 = pq
p = 1728/q
Substituting this into the first equation, we get:
3(24/q) = 1728/q
72 = 1728/q^2
q^2 = 24^2
Since q is a positive number, we take the positive square root:
q = 24
Substituting this into the equation for p, we get:
p = 1728/24 = 72
Therefore, the value of p is 72 and the value of q is 24.
An exponential sequence is a sequence in which each term is the product of the previous term and a constant factor. We can write the given terms as:
3, 3 * r, 3 * r^2, 24
where r is the constant factor.
Now, we can set up a relationship between consecutive terms:
3 * r = p
3 * r^2 = q
3 * r^3 = 24
Solve for r from the third equation:
r^3 = 8
r = 2 (since 2^3 = 8)
Now we can find p and q:
p = 3 * r = 3 * 2 = 6
q = 3 * r^2 = 3 * (2^2) = 3 * 4 = 12
Hence, the values of p and q are 6 and 12 respectively.