Question

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.

Answer 1

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