Advertisements
Advertisements
Question
If a1, a2, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an) is equal to cot a1 – cot an
Advertisements
Solution
We have sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an)
= `sin d[1/(sina_1 sina_2) + 1/(sina_2 sina_3) + ... + 1/(sina_(n - 1) sina_n)]`
= `(sin(a_2 - a_1))/(sina_1 sina_2) + (sin(a_3 - a_2))/(sina_2 sina_3) + ... + (sin(a_n - a_(n - 1)))/(sina_(n - 1) sina_n)`
= `(sina_2 cos a_1 - cosa_2 sina_1)/(sina_1 sina_2) + (sina_3 cosa_2 - cosa_3 sina_2)/(sina_2 sina_3) + ... + (sina_n cosa_(n - 1) - cosa_n sina_(n - 1))/(sina_(n - 1) sina_n)`
= (cot a1 – cot a2) + (cot a2 – cot a3) + ... + (cot an–1 – cot an)
= cot a1 – cot an
APPEARS IN
RELATED QUESTIONS
In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
Find the sum to n terms of the A.P., whose kth term is 5k + 1.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
−1, 1/4, 3/2, 11/4, ...
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\]
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
Find the 12th term from the following arithmetic progression:
3, 8, 13, ..., 253
The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.
Find the sum of the following arithmetic progression :
a + b, a − b, a − 3b, ... to 22 terms
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Find the sum of all integers between 100 and 550, which are divisible by 9.
The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
In an A.P. the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that 20th term is −112.
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) are in A.P.
If a2, b2, c2 are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.
If a, b, c is in A.P., then show that:
a2 (b + c), b2 (c + a), c2 (a + b) are also in A.P.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.
A man starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?
Write the common difference of an A.P. whose nth term is xn + y.
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
If a, b, c are in A.P. and x, y, z are in G.P., then the value of xb − c yc − a za − b is
Find the sum of first 24 terms of the A.P. a1, a2, a3, ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad
Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
If 100 times the 100th term of an A.P. with non zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is ______.
