If a1,a2,a3,............an  are in A.P and1a1an+1a2an−1+1a3an*2+.......+1ana1=K(1a1+1a2+1a3+..................+1an) . Then K is

We know that in an A.P a1+an=a2+an−1=a3+an−2=.............(i) [see the properties of A.P]∴1a1an+1a2an−1+1a3an−2+..........+1ana1 =1a1+an[a1+ana1an+a1+ana2an−2+a1+ana3an−2+..........+a1+anana1] =1a1+an[a1+ana1an+a1+an−1a2an−2+a1+an−2a3an−2+..........+an+an1ana1]       [using (i)]=1a1+an[(1a1+1an)+(1a2+1an−1)+(1a3+1an−2)+..........+(1a1+1an)] =2a1+an[1a1+1a2+1a3+.........+1an] (each term occurs twice).