Let `A_1,A_2,....A_n` be the vertices of an n-sided regular polygon such that ,`1/(A_1A_2)=1/(A_1A_3)+1/(A_1A_4)`. Find the value of n.

A_1,A_2,..., A_n are the vertices of a regular plane polygon with n sides and O as its centre. Show that sum_(i=1)^n vec (OA)_i xx vec(OA)_(i+1)=(1-n)(vec (OA)_2 xx vec(OA)_1)

A_(1),A_(2), …. A_(n) are the vertices of a regular plane polygon with n sides and O ars its centre. Show that sum_(i=1)^(n-1) (vec(OA_(i))xxvec(OA)_(i+1))=(1-n) (vec(OA)_2 xx vec(OA)_(1))

If "^(2n+1)P_(n-1):^(2n-1)P_n=3:5, then find the value of ndot

Let {a_n}(ngeq1) be a sequence such that a_1=1,a n d3a_(n+1)-3a_n=1 for all ngeq1. Then find the value of a_(2002.)

Let T_(n) denote the number of triangles which can be formed by using the vertices of a regular polygon of n sides. If T_(n + 1) - T_(n) = 36 , then n is equal to

If the equation of the locus of a point equidistant from the points (a_1, b_1) and (a_2, b_2) is (a_1-a_2)x+(b_1-b_2)y+c=0 , then the value of c is

Let a_1,a_2,a_3 ….and b_1 , b_2 , b_3 … be two geometric progressions with a_1= 2 sqrt(3) and b_1= 52/9 sqrt(3) If 3a_99b_99=104 then find the value of a_1 b_1+ a_2 b_2+…+a_nb_n

let a_1,a_2,...a_n be in A.P. wth common difference pi/6 . if seca_1seca_2+seca_2seca_3+....seca_(n-1)seca_n = k(tana_n-tana_1) Find the value of k

If a_1,a_2, a_3, a_4 be the coefficient of four consecutive terms in the expansion of (1+x)^n , then prove that: (a_1)/(a_1+a_2)+(a_3)/(a_3+a_4)=(2a_2)/(a_2+a_3)dot

Let a_1,a_2,.........a_n be real numbers such that sqrt(a_1)+sqrt(a_2-1)+sqrt(a_3-2)++sqrt(a_n-(n-1))=1/2(a_1+a_2+.......+a_n)-(n(n-3)/4 then find the value of sum_(i=1)^100 a_i