If 3,p,q,24,are Consecutive Terms Of An Exponential Sequence,find The Values Of P And Q
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Finish the solving
Let the common ratio be r then we have
3*r=p
P*r=q
Q*r=24
Divide the second equation by the first
P*r/3*r=q/p
P/3=q/p
P*2 =3q
From the third equation, we have:
q = 24/r
Substituting into the last equation yields:
P*2 = 3(24/r)
P*2 = 72/r
Multiplying both sides by 3r gives:
3Pr*2 = 216
Dividing both sides by 3P gives:
r*2 = 72/3P
r*2 = 24/P
Substituting into the second equation gives:
P*q/P = 24
q = 24
Substituting into the first equation gives:
3*r = p
In summary,
p = 6
q = 24
To find the values of p and q in the given exponential sequence, we need to determine the common ratio first.
In an exponential sequence, each term is obtained by multiplying the previous term by a constant ratio. Let's consider the given terms:
3, p, q, 24
To find the common ratio (r), we can divide any term by the previous term. For example, dividing p by 3 would give us:
p/3 = r
Now, let's consider the next term q. We can divide q by p to get:
q/p = r
Since both expressions equal r, we can equate them:
p/3 = q/p
To simplify this equation, we can cross multiply:
p^2 = 3q
Rearranging the equation, we have:
3q = p^2
Now, let's use another pair of consecutive terms, q and 24. We can divide 24 by q to get:
24/q = r
Substituting r with p/3, we have:
24/q = p/3
We can now cross multiply:
3q = 24p
Rearranging the equation, we have:
24p = 3q
We now have a system of equations:
1) 3q = p^2
2) 24p = 3q
To solve this system, we can substitute the value of 3q from equation 1 into equation 2:
24p = 3(p^2)
Dividing both sides by 3:
8p = p^2
Rearranging the equation:
p^2 - 8p = 0
Factoring out p:
p(p - 8) = 0
From this equation, we can see that p = 0 or p = 8. However, if p = 0, then q would also equal 0, resulting in a constant sequence. Hence, we can conclude that p = 8.
Substituting p = 8 into equation 1:
3q = 8^2
3q = 64
q = 64/3
Therefore, the values of p and q in the given exponential sequence are p = 8 and q = 64/3.
24 = 3r^3
r^3 = 24/3 = 8
r = 2
so, ...