The following gives a geometic sequence 2, P Q 250 find the values of P and Q
THE FOLLOWING GIVE A GEOMETRIC SEQUENCE: 2, p, q 250 FIND THE VALUE OF P AND Q
Thank you very much
We know that given a ratio of r,
P = 2r
Q = 2r²
250 = 2r³, so r=5
P=10
Q=50
Do the right solving
To find the values of P and Q in the geometric sequence 2, P, Q, 250, we can use the formula for a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a constant ratio.
Let's identify the ratio between the terms. We divide any term by its previous term to get the common ratio:
P/2 = Q/P = 250/Q
From the first two ratios, we can deduce that P^2 = 2Q.
Now, let's look at the third ratio: 250/Q. Since the common ratio applies to all terms, we know that Q/(250/Q) = (Q^2)/250.
Since Q/(250/Q) = Q^2/250, we can set these equal to each other:
Q^2/250 = (Q^2)/250
This implies that Q^2 = 250.
Therefore, the value of Q can be found by taking the square root of 250. In this case, Q has two possible values, positive and negative:
Q = ±√(250) = ± 5√(10).
Now, let's substitute the value of Q back into the equation P^2 = 2Q:
P^2 = 2(± 5√(10)),
which simplifies to:
P^2 = ± 10√(10).
Therefore, the values of P can be found by taking the square root of ± 10√(10). Again, we have two possible values, positive and negative:
P = ±√(± 10√(10)).
In summary, the values of P and Q in the geometric sequence 2, P, Q, 250 are:
Q = ± 5√(10),
P = ±√(± 10√(10)).