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.

Answer 1

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.