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If a1,a2,a3, ,an are in A.P. with common...

If `a_1,a_2,a_3, ,a_n` are in A.P. with common difference `d(w h e r ed!=0)`, then the sum of series. `sind (coseca_1 coseca_2+coseca_2 coseca_3 +....+ coseca_(n-1) cos e ca_n)` is equal to `cot a_1-cot a_n`

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To solve the problem, we need to prove that the sum of the series \[ \sin d \left( \csc a_1 \csc a_2 + \csc a_2 \csc a_3 + \ldots + \csc a_{n-1} \csc a_n \right) \] is equal to ...
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If a_1, a_2, a_3,...a_n are in A.P with common difference d !=0 then the value of sind(coseca_1 coseca_2 +cosec a_2 cosec a_3+...+cosec a_(n-1) cosec a_n) will be

If a_1, a_2, a_n are in A.P. with common difference d!=0 , then the sum of the series sind[seca_1seca_2+(sec)_2seca_3+....+s e ca_(n-1)(sec)_n] is : a.cos e ca_n-cos e ca b. cota_n-cota c. s e ca_n-s e ca d. t a na_n-t a na

If a_(1),a-(2),a_(3),………….a_(n) are in A.P. with common differecne d, prove that sin [cosec a_(1)coseca_(2)+cosec a_(2)cosec a_(3)+………….+cosec a_(n-1)cosec a_(n)]=cota_(1)-cota_(n) .

4.If a, a, ag.....an are in A.P. and commondifference is d then the value ofsind(coseca, coseca.y + coseca, cosecaz+ ... cosecdn-1 cosecan)will be-(1) seca1 - Secon (2) coseca1 - cosecan(3) cota1-cotan(4) tandi-tanan

If a_1,a_2,a_3,.....,a_n are in AP where a_i ne kpi for all i , prove that cosec a_1* cosec a_2+ cosec a_2* cosec a_3+...+ cosec a_(n-1)* cosec a_n=(cota_1-cota_n)/(sin(a_2-a_1)) .

If a_1,a_2,a_3,...,a_n be in AP whose common difference is d then prove that sum_(i=1)^n a_ia_(i+1)=n{a_1^2+na_1d+(n^2-1)/3 d^2} .

If a_1,a_2,a_3,…………..a_n are in A.P. whose common difference is d, show tht sum_2^ntan^-1 d/(1+a_(n-1)a_n)= tan^-1 ((a_n-a_1)/(1+a_na_n))

RD SHARMA-ARITHMETIC PROGRESSIONS-Solved Examples And Exercises
  1. If theta1,theta2,theta3, ,thetan are in AP, whose common difference ...

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  2. If (m+1) term of an A.P. is twice the(n+1) th term, prove that (3m+1) ...

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  3. If a1,a2,a3, ,an are in A.P. with common difference d(w h e r ed!=0), ...

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  4. If an A.P. is such that (a4)/(a7)=2/7, find (a6)/(a8) .

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  5. The 4th term of an A.P. is three times the first and the 7th term e...

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  6. Find the number of terms common to the two AP's 3,7,11,15.... 407 and...

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  7. Show that in an A.P. the sum of the terms equidistant from the begi...

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  8. If a1,a2,a3, b, ... ,an are an A.P. of non-zero terms, prove that 1/...

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  9. If a1,a2,a3, ,an are in A.P., where ai >0 for all i , show that 1/(sqr...

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  10. The p^(t h) term of an A.P. is a and q^(t h) term is b Prove that t...

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  11. If S1,S2, S3, Sm are the sums of n terms of m A.P. ' s whose first ter...

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  12. Let Sn be the sum of first n terms of an A.P. with non-zero common d...

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  13. If there are (2n+1) terms in A.P., then prove that the ratio of the su...

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  14. The ratio of the sum of n terms of two A.P. s is (7n+1):(4n+27) . Fin...

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  15. Show that x^2+x y+y^2,z^2+xz+x^2, y^2+y z+z^2, are consecutive terms o...

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  16. If a ,b ,c are in A.P., prove that: i.(a-c)^2=4(a-b)(b-c) ii.a^2+c^2...

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  17. Find the sum of all those integers between 100 and 800 each of which o...

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  18. If the sum of m terms of an A.P. is equal to the sum of either the nex...

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  19. Suppose xa n dy are two real numbers such that the rth mean between xa...

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  20. The sum of two numbers is (13)/6dot An even number of arithmetic means...

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