The numbers P,10,and Q are three consecutive term of an Arithmetic progression, The numbers P,6,and Q are three consecutive term of a GP .Form a quadratic equation with respect to Q, satisfying the series.
Since P, 10, and Q are three consecutive terms of an arithmetic progression, we have that Q - 10 = 10 - P. Simplifying this equation gives Q = P + 20.
Since P, 6, and Q are three consecutive terms of a geometric progression, we have that Q/6 = 6/P. Simplifying this equation gives Q = 36/P.
Substituting the value of Q from the first equation into the second equation gives 36/P = P + 20. Multiplying both sides by P gives 36 = P^2 + 20P.
Rearranging this equation gives P^2 + 20P - 36 = 0.
So the quadratic equation with respect to Q that satisfies the given series is Q = P + 20, or more specifically, Q^2 + 20Q - 36 = 0.
To form a quadratic equation with respect to Q, let's first find the common difference (d) of the arithmetic progression.
We know that the three consecutive terms of the arithmetic progression are P, 10, and Q.
The second term (10) is given by:
10 = P + d
We also know that the three consecutive terms of the geometric progression are P, 6, and Q.
The second term (6) is given by:
6 = P × r
Now, we can use these two equations to form the quadratic equation.
From the first equation, we can rewrite it as:
d = 10 - P
Since we know that the second term of the arithmetic progression is Q, we can rewrite it as:
Q = P + d
Substituting the value of d from the first equation into the second equation, we get:
Q = P + (10 - P)
Simplifying further, we have:
Q = 10
Therefore, the quadratic equation with respect to Q satisfying the series is:
Q - 10 = 0
To form the quadratic equation, we need to find the common difference (d) for the arithmetic progression (AP) and the common ratio (r) for the geometric progression (GP).
Let's find the common difference (d) of the arithmetic progression first:
In an arithmetic progression, the difference between consecutive terms is constant. So, we have:
P + d = 10
10 + d = Q
Subtracting the first equation from the second equation, we get:
(Q - 10) - (10 - P) = d
Q - 10 - 10 + P = d
Q + P - 20 = d
Next, let's find the common ratio (r) of the geometric progression:
In a geometric progression, the ratio of any two consecutive terms is constant. So, we have:
P * r = 6
6 * r = Q
Dividing the second equation by the first equation, we get:
(Q / 6) / (P / 6) = r
Q / P = r
Now, we can form the quadratic equation with respect to Q:
Since we have Q + P - 20 = d and Q / P = r, we substitute these values into the equation Q + P - 20 = d:
(Q / P) + P - 20 = d
Substituting Q / P = r, we get:
r + P - 20 = d
Finally, since d = Q + P - 20, we substitute this value into the equation:
r + P - 20 = Q + P - 20
Simplifying, we get the quadratic equation:
Q - r - Q = 0
Therefore, the quadratic equation with respect to Q, satisfying the series, is:
Q^2 - rQ = 0