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nth Term of A.P

If a1, a2, ...

Question

If $a_{1}, a_{2}, a_{3}, . . . ., a_{n - 1}, a_{n}$ are in A.P., then show that $\frac{1}{a_{1} a_{n}} + \frac{1}{a_{2} a_{n - 1}} + \frac{1}{a_{3} a_{n - 2}} + . . . . . \frac{1}{a_{n} a_{1}} = \frac{2}{(a_{1} + a_{n})} [\frac{1}{a_{1}} + \frac{1}{a_{2}} + . . . . \frac{1}{a_{n}}]$ .

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a_{1}, a_{2}, a_{3}, . . . . . . a_{n}

\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}} = \frac{2}{a_{1} + a_{n}} (\frac{1}{a_{1}} + \frac{1}{a_{2}} + . . . + \frac{1}{a_{n}}) .

\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}} = \frac{2}{a_{1} + a_{n}} (\frac{1}{a_{1}} + \frac{1}{a_{2}} + . . . + \frac{1}{a_{n}})

a_{1}, a_{2}, a_{3}, . . . . ., a_{n}

\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}} = \frac{2}{a_{1} + a_{n}} (\frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + . . . + \frac{1}{a_{n}})

a_{1} + a_{n} = a_{r} + a_{n - r}

1 \leq r \leq n

a_{1}, a_{2}, a_{3} . . . . . a_{n}

\frac{1}{a_{1} a_{n}} + \frac{1}{a_{2} a_{a - 1}} + \frac{1}{a_{3} a_{n - 2}} + . . . . . + \frac{1}{a_{n} a_{1}} =

a_{1}, a_{2}, a_{3}, . . ., a_{n}

a_{i} > 0

\frac{1}{a_{1} a_{2}} + \frac{1}{a_{2} a_{3}} + . . . + \frac{1}{a_{n - 1} a_{n}} = \frac{n - 1}{a_{1} a_{n}}

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