Question
If $${a_1},{a_2},{a_3},.....,{a_n}$$ are in A.P. and $$\frac{1}{{{a_1}{a_n}}} + \frac{1}{{{a_2}{a_{n - 1}}}} + \frac{1}{{{a_3}{a_{n - 2}}}} + ..... + \frac{1}{{{a_n}{a_1}}} = K\left( {\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \frac{1}{{{a_3}}} + ..... + \frac{1}{{{a_n}}}} \right).{\text{ Then }}K{\text{ is}}$$
A.
$$\frac{2}{{{a_1} + {a_n}}}$$
B.
$$\frac{n}{{{a_1} + {a_n}}}$$
C.
$$\frac{1}{{{a_1} + {a_n}}}$$
D.
$$\frac{n - 1}{{{a_1} + {a_n}}}$$
Answer :
$$\frac{2}{{{a_1} + {a_n}}}$$
Solution :
We know that in an A.P.
$${a_1} + {a_n} = {a_2} + {a_{n - 1}} = {a_3} + {a_{n - 2}}\,\,\,\,.....\left( {\text{i}} \right)$$
[see the properties of A.P.]
$$\eqalign{
& \therefore \frac{1}{{{a_1}{a_n}}} + \frac{1}{{{a_2}{a_{n - 1}}}} + \frac{1}{{{a_3}{a_{n - 2}}}} + ..... + \frac{1}{{{a_n}{a_1}}} \cr
& = \frac{1}{{{a_1} + {a_n}}}\left[ {\frac{{{a_1} + {a_n}}}{{{a_1}{a_n}}} + \frac{{{a_1} + {a_n}}}{{{a_2}{a_{n - 2}}}} + \frac{{{a_1} + {a_n}}}{{{a_3}{a_{n - 2}}}} + ..... + \frac{{{a_1} + {a_n}}}{{{a_n}{a_1}}}} \right] \cr
& = \frac{2}{{{a_1} + {a_n}}}\left[ {\frac{1}{{{a_1}}} + \frac{1}{{{a_2}}} + \frac{1}{{{a_3}}} + ..... + \frac{1}{{{a_n}}}} \right] \cr} $$