The numbers P,10,and Q are three consecutive term of an Arithmetic progression, The number's P,6,and Q are three consecutive term of a GP .Form a quadratic equation with respect to Q, satisfying the series.
Let's start by assuming that the common difference of the arithmetic progression is d.
Since P, 10, and Q are consecutive terms of the arithmetic progression, we can write:
P = 10 - d
Q = 10 + d
Similarly, assuming that the common ratio of the geometric progression is r, we can write:
P = 6 / r
Q = 6
To form a quadratic equation with respect to Q, we can equate the expressions for Q from both progressions:
10 + d = 6
Simplifying this equation, we have:
d = 6 - 10
d = -4
Substituting the value of d into the expressions for P and Q in the arithmetic progression:
P = 10 - (-4) = 10 + 4 = 14
Q = 10 + (-4) = 10 - 4 = 6
Therefore, the quadratic equation with respect to Q that satisfies the series is:
Q^2 - 20Q + 84 = 0
To form a quadratic equation with respect to Q, let's first determine the common difference (d) of the Arithmetic Progression (AP) and the common ratio (r) of the Geometric Progression (GP).
Given: P, 10, Q are three consecutive terms of an Arithmetic Progression.
So, the common difference (d) can be found by subtracting the second term from the first term:
d = 10 - P
Also given: P, 6, Q are three consecutive terms of a Geometric Progression.
So, the common ratio (r) can be found by dividing the second term by the first term:
r = 6 / P
Now, let's write the equation for the terms of the AP and the GP.
For the AP:
P = P (1st term)
10 = P + d (2nd term)
Q = P + 2d (3rd term)
For the GP:
P = P (1st term)
6 = P * r (2nd term)
Q = P * r^2 (3rd term)
Since we need to form a quadratic equation with respect to Q, we can use the third term of the GP expression:
Q = P * r^2
Substituting the values of r and d into the equation:
Q = P * (6 / P)^2
Q = 6^2
Q = 36
Therefore, the quadratic equation with respect to Q is:
Q = 36
To form a quadratic equation with respect to Q satisfying the given series, we need to find the common difference (d) of the Arithmetic Progression (AP) and the common ratio (r) of the Geometric Progression (GP).
Let's start by finding the common difference (d) of the Arithmetic Progression (AP):
Since P, 10, and Q are consecutive terms of an AP, we can express them as:
P = a + (n-1)d
10 = a + nd
Q = a + (n+1)d
Here, a represents the first term, d represents the common difference, and n represents the position of the terms in the series.
Now, let's find the common ratio (r) of the Geometric Progression (GP) using the values P, 6, and Q:
P = ar^(n-1)
6 = ar^n
Q = ar^(n+1)
To find r, divide the second equation by the first equation:
(6/ar^n) = (ar^n)/(ar^(n-1))
6 = r
Now that we have r = 6, we can substitute this value into the equations:
P = a*6^(n-1)
10 = a*6^n
Q = a*6^(n+1)
Finally, we can form a quadratic equation with respect to Q:
Q = a*6^(n+1)
Q = a*6^1*6^n
Q = 6a*6^n
The quadratic equation is:
Q^2 - 36a*Q = 0
So, the quadratic equation with respect to Q, satisfying the given series, is Q^2 - 36a*Q = 0.