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. Prove or disprove the converse of this result.

Let A_(1),A_(2),...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.

let A_(1),A_(2),A_(3),...A_(n) are the vertices of a regular n sided polygon inscribed in a circle of radius R.If (A_(1)A_(2))^(2)+(A_(1)A_(3))^(2)+...(A_(1)A_(n))^(2)=14R^(2) then find the number of sides in the polygon.

If a_(1),a_(2),a_(3),,a_(n) are an A.P.of non-zero terms, prove that _(1)(1)/(a_(1)a_(2))+(1)/(a_(2)a_(3))++(1)/(a_(n-1)a_(n))=(n-1)/(a_(1)a_(n))

If a_(1), a_(2), a_(3),…, a_(n) be in A.P. Show that, (1)/(a_(1)a_(2)) + (1)/(a_(2)a_(3)) +….+(1)/(a_(n-1)a_(n)) = (n-1)/(a_(1)a_(n))

If a_(1), a_(2) , a_(3),…,a _(n+1) are in A. P. then the value of (1)/(a _(1)a_(2))+(1)/(a_(2)a_(3))+(1)/(a_(3)a_(4))+...+(1)/(a_(n)a_(n+1)) is-

If a_(1), a_(2), a_(3) ,…., a_(n) are the terms of arithmatic progression then prove that (1)/(a_(1)a_(2)) + (1)/(a_(2)a_(3)) + (1)/(a_(3)a_(4)) + ….+ (1)/(a_(n-1) a_(n)) = (n-1)/(a_(1)a_(n))

If 1,a_(1),a_(2),...,a_(n-1) are the nth roots of unity then prove that (1-a_(1))(1-a_(2))(1-a_(3))=(1-a_(n-1))=n

Let a_(1),a_(2),a_(3),...a_(n) be an AP.Prove that: (1)/(a_(1)a_(n))+(1)/(a_(2)a_(n-1))+(1)/(a_(3)a_(n-2))+......+(1)/(a_(n)a_(1))=

If a_(1), a_(2), a_(3), …., a_(n) are in H.P., prove that, a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) +…+ a_(n-1)a_(n) = (n-1)a_(1)a_(n)