x,y and z are in A.P. `A_(1)` is the A.M. of x and y. `A_(2)` is the A.M of y and z. Prove that the A.M. of `A_(1) and A_(2)` is y.

If a,b,c are in G.P and A.M of a and b is x and A.M. of b and c is y then…….

Insert for A.M's between (1)/(2) and 3 and prove that A_(1) + A_(2) + A_(3) + A_(4)= 7

A mass M attached to a horizontal spring executes SHM with an amplitude A_(1) . When mass M passes through its mean position a smaller mass m is placed over it and both of them move togther with amplitude A_(2) . Ratio of ((A_(1))/(A_(2))) is:

If A_(1),A_(2),A_(3),…,A_(n) are n points in a plane whose coordinates are (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)),…,(x_(n),y_(n)) respectively. A_(1)A_(2) is bisected in the point G_(1) : G_(1)A_(3) is divided at G_(2) in the ratio 1 : 2, G_(2)A_(4) at G_(3) in the1 : 3 and so on untill all the points are exhausted. Show that the coordinates of the final point so obtained are (x_(1)+x_(2)+.....+ x_(n))/(n) and (y_(1)+y_(2)+.....+ y_(n))/(n)

Path different between waves y_(1) = A_(1) sin (omega t - (2pix)/(lamda)) and y _(2) = A_(2) cos (omega t - (2pi x)/(lamda) + phi) at the point of superpositon is :

If x, y, z are in A.P. and tan^(-1)x , tan^(-1)y and t a n^(-1)z are also in A.P., then (1) 2x""=""3y""=""6z (2) 6x""=""3y""=""2z (3) 6x""=""4y""=""3z (4) x""=""y""=""z

Find the sum of first 24 terms of the A.P. a_(1) , a_(2), a_(3) ...., if it is know that a_(1) + a_(5) + a_(10) + a_(15) + a_(20) + a_(24) = 225

The number of ways in which 10 condidates A_(1),A_(2),......,A_(10) can be ranked so that A_(1) is always above A_(2) , is

If y= (x + sqrt(x^(2) + 1))^(m) then prove that, (x^(2)+1) y_(2) + xy_(1)= m^(2)y

If a_(1), a_(2), a_(3).,,,,,,,,a_(n) are in A.P and their common difference is d. The value of the series sin d_(1) [sec a_(1).sec a_(2) + sec a_(2).sec a_(3)+ ….+ sec a_(n-1).sec a_(n)] is……..