Question

If a_1,a_2,a_3 .....a_n be in AP Prove 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)]

Answer 1

Can you latexify it? :)

Answer 2

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Answer 3

in RHS
in 1/a1 + 1/a2 + 1/a3 ........ 1/an-2 + 1/an-1 + 1/an
group those terms which sum to 1+n i.e
(1/a1 + 1/an) + (1/a2 + 1/an-1) + (1/a3 + 1/an-2) ...
=> (an + a1)/a1an + (a2 + an-1)/a2an-1 + (a3 + an-2)/a3an-2
now an= a1 + nd - d
an-1 = a1 + nd - 2d and a2 = a1 + d
so an-1 + a2 = a1 + an
similarily all the numerators are equal
so we have this :
(an + a1) (1/a1an + 1/a2an-1 + ...... 1/a(n/2 -1) a (n/2 +1) )
mutiply divide this by 2
[(an + a1)/2 ] (2/a1an + 2/a2an-1 + ...... 2/a(n/2 -1) a (n/2 +1) )
[(an + a1)/2] (1/a1an + 1/a2an-1 + ...... 1/ana1)
now RHS was multiplied by (2/(an +a1))
so on doing this,, we come to LHS..
hence proved..
sorry for the rigorous writing,,was i clear?