Given four points `P_1,P_2,P_3a n dP_4` on the coordinate plane with origin `O` which satisfy the condition `( vec (O P))_(n-1)+( vec(O P))_(n+1)=3/2 vec (O P)_(n)`. If P1 and P2 lie on the curve xy=1 , then prove that P3 does not lie on the curve

IF P_1, P_2, P_3, P_4 are points in a plane or space and O is the origin of vectors, show that P_4 coincides with Oiff( vec O P)_1+ vec P_1P_2+ vec P_2P_3+ vec P_3P_4= vec 0.

Find n if : ""^(2n+1) P_(n-1) : ""^(2n-1) P_n = 3: 5

If P(n) is the statement n(n+1) is even, then what is P(3)?

If P_m stands for ^m P_m , then prove that: 1+1. P_1+2. P_2+3. P_3++ndotP_n=(n+1)!

The coordinates of the point P are (-3,\ 2) . Find the coordinates of the point Q which lies on the line joining P and origin such that O P=O Q .

If P(n) is the statement 2^ngeq3n , and if P(r) is true, prove that P(r+1) is true.

Let vec r_1, vec r_2, vec r_3, , vec r_n be the position vectors of points P_1,P_2, P_3 ,P_n relative to the origin Odot If the vector equation a_1 vec r_1+a_2 vec r_2++a_n vec r_n=0 hold, then a similar equation will also hold w.r.t. to any other origin provided a. a_1+a_2+dot+a_n=n b. a_1+a_2+dot+a_n=1 c. a_1+a_2+dot+a_n=0 d. a_1=a_2=a_3dot+a_n=0

If P(n) be the statement " 10n+3 is a prime number", then prove that P(1) and P(2) are true but P(3) is false.

If A( vec a),B( vec b)a n dC( vec c) are three non-collinear points and origin does not lie in the plane of the points A ,Ba n dC , then point P( vec p) in the plane of the A B C such that vector vec O P is _|_ to planeof A B C , show that vec O P=([ vec a vec b vec c]( vec axx vec b+ vec bxx vec c+ vec cxx vec a))/(4^2),w h e r e is the area of the A B Cdot

If vec P O+ vec O Q= vec Q O+ vec O R , show that the point, P ,Q ,R are collinear.