Let `A_(1), A_(2), A_(3),…,A_(n)` be the vertices of an n-sided regular polygon such that `(1)/(A_(1)A_(2))=(1)/(A_(1)A_(3))+(1)/(A_(1)A_(4)).` Find the value of n.

If A_(1), A_(2),..,A_(n) are any n events, then

Let A_(1),A_(2),A_(3),...A_(12) are vertices of a regular dodecagon. If radius of its circumcircle is 1, then the length A_(1)A_(3) is-

Let a_(1),a_(2),a_(3),….,a_(4001) is an A.P. such that (1)/(a_(1)a_(2))+(1)/(a_(2)a_(3))+...+(1)/(a_(4000)a_(4001))=10 a_(2)+a_(4000)=50 . Then |a_(1)-a_(4001)| is equal to

If a_(1), a_(2), . . . , a_(n) is a sequence of non-zero number which are in A.P., show that (1)/(a_(1)a_(n))+(1)/(a_(2)a_(n-1))+. . . + (1)/(a_(n)a_(1))= (2)/(a_(1) + a_(n)) [(1)/(a_(1))+(1)/(a_(2))+. . . +(1)/(a_(n))]

Let a_(1),a_(2)…,a_(n) be a non-negative real numbers such that a_(1)+a_(2)+…+a_(n)=m and let S=sum_(iltj) a_(i)a_(j) , then

If A_(1), A_(2), A_(3),....A_(51) are arithmetic means inserted between the number a and b, then find the value of ((b + A_(51))/(b - A_(51))) - ((A_(1) + a)/(A_(1) - a))

If a_(1), a_(2), a_(3) are in arithmetic progression and d is the common diference, then tan^(-1)((d)/(1+a_(1)a_(2)))+tan^(-1)((d)/(1+a_(2)a_(3)))=

a_(1),a_(2),a_(3),a_(4),a_(5), are first five terms of an A.P. such that a_(1) +a_(3) +a_(5) = -12 and a_(1) .a_(2) . a_(3) =8 . Find the first term and the common difference.

Let A_(1)A_(2)A_(3)………………. A_(14) be a regular polygon with 14 sides inscribed in a circle of radius 7 cm. Then the value of (A_(1)A_(3))^(2) +(A_(1)A_(7))^(2) + (A_(3)A_(7))^(2) (in square cm) is……………..

A squence of positive terms A_(1),A_(2),A_(3),"....,"A_(n) satisfirs the relation A_(n+1)=(3(1+A_(n)))/((3+A_(n))) . Least integeral value of A_(1) for which the sequence is decreasing can be