Question

If A1, A2, A3 ,............... An-1, An are in an A.P then show that 1/A1An + 1/A2 An-1 + 1/A3 An-2.......... +1/AnA1 = 2/A1 + An ( 1/A1 + 1/A2+............1/An).

Answer 1

A₁ , A₂ , A₃ , A₄ .....An are in AP
so, $A_1+A_n=A_2+A_{n-1}=A_3+A_{n-2}=.......= P[ \text{Let}]$
we can write,
$\frac{1}{A_1.A_n}=\frac{1}{A_1+A_n}.[\frac{1}{A_1}+\frac{1}{A_n}]=\frac{1}{K}[\frac{1}{A_1}+\frac{1}{A_n}]...............(1)$
$\frac{1}{A_2.A_{n-2}}=\frac{1}{A_2+A_{n-2}}.[\frac{1}{A_2}+\frac{1}{A_{n-2}}]=\frac{1}{K}[\frac{1}{A_2}+\frac{1}{A_{n_2}}]...............(2)$
similarly,
.................
..........................
$\frac{1}{A_n.A_1}=\frac{1}{A_n+A_1}.[\frac{1}{A_n}+\frac{1}{A_1}]=\frac{1}{K}[\frac{1}{A_n}+\frac{1}{A_1}]...............(n)$

add all equations ,
Then,
$\frac{1}{A_1.A_n}+\frac{1}{A_2.A_{n-2}}+\frac{1}{A_3.A_{n-2}}+..........+\frac{1}{A_n.A_1}=\frac{1}{K}[\frac{1}{A_1}+\frac{1}{A_n}]+.....$
= 1/K[1/A₁ + 1/An ] + 1/K[1/A₂+ 1/Aₙ₋₁ ] + ............+ 1/K[ 1/An + 1/A₁ ]
= 2/K[ 1/A₁ + 1/A₂ + 1/A₃ + 1/A₄ + ........+ 1/An]

Now, put K = (A₁ + An)
Then, = 2/(A₁ + An) [ 1/A₁ + 1/A₂ + 1/A₃ + 1/A₄ + ....... + 1/An ]
Hence, LHS = RHS