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Arithmetic Mean

Let a 1, a 2,...

Question

Let $a_{1}, a_{2}, a_{3}, . . .$ be an A.P. with common difference $π / 6$ and $cosec a_{1} cosec a_{2} + cosec a_{2} cosec a_{3} + . . . cosec a_{n - 1} cosec a_{n} = k (cot a_{1} - cot a_{n})$ . Then $k$

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Solution

$cosec a_{1} cosec a_{2} + cosec a_{2} cosec a_{3} + . . . cosec a_{n - 1} cosec a_{n}$
$= \frac{1}{sin a_{1} sin a_{2}} + \frac{1}{sin a_{2} sin a_{3}} + . . . + \frac{1}{sin a_{n - 1} sin a_{n}}$
$= \frac{1}{sin π / 6} [\frac{sin (a_{2} - a_{1})}{sin a_{1} sin a_{2}} + \frac{sin (a_{3} - a_{2})}{sin a_{2} sin a_{3}} + . . . + \frac{sin (a_{n} - a_{n - 1})}{sin a_{n - 1} sin a_{n}}]$
$= 2 [(cot a_{1} - cot a_{2}) + (cot a_{2} - cot a_{3}) + . . . + (cot a_{n - 1} - cot a_{n})]$
$= 2 [(cot a_{1} - cot a_{2})]$

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Similar questions

If $a_{1}, a_{2}, a_{3}, \dots \dots a_{n}$ are in A.P., with common difference d, then the sum of the series $s i n d [c o s e c a_{1} c o s e c a_{2} + c o s e c a_{1} c o s e c a_{3} + \dots \dots + c o s e c a_{n} - 1 c o s e c a_{n}]$ is

Q. If a₁, a₂, a₃, .... a_n are in A.P. with common difference d, then the sum of the series sin d [cosec a₁ cosec a₂ + cosec a₁ cosec a₃ + .... + cosec a_n_{− 1} cosec a_n] is
(a) sec a₁ − sec a_n
(b) cosec a₁ − cosec a_n
(c) cot a₁ − cot a_n
(d) tan a₁ − tan a_n

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

be an A.P. with common difference

\frac{π}{6}

sec a_{1} sec a_{2} + sec a_{2} sec a_{3} + . . . . + sec a_{n - 1} sec a_{n} . = k (tan a_{n} - tan a_{1})

find the value of

k

.

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

a_{6} = 2

. Then the common difference of this A.P., which maximises the produce

a_{1} a_{4} a_{5}

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

be an A.P. with

a_{6} = 2.

Then the common difference of this A.P., which maximises the product

a_{1} a_{4} a_{5},

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Arithmetic Mean

Standard XII Mathematics

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