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If `a_1,a_2,a_3, ,a_n` are in A.P., where `a_i >0` for all `i` , show that `1/(sqrt(a_1)+sqrt(a_2))+1/(sqrt(a_2)+sqrt(a_3))+.............+1/(sqrt(a_(n-1))+sqrt(a_n))=(n-1)/(sqrt(a_1)+sqrt(a_n))dot`

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If a_1,a_2,a_3,………a_n are in A.P, where a_igt0 for all i show that 1/(sqrt(a_1)+sqrt(a_2))+1/(sqrt(a_2)+sqrt(a_3))+……..+1/(sqrt(a_(n-1))+sqrt(a_n))=(n-1)/(sqrt(a_1)+sqrt(a_n))

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 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))

If a_1,a_2,a_3,.....,a_n are in AP, prove that 1/(a_1a_2)+1/(a_2a_3)+1/(a_3a_4)+...+1/(a_(n-1)a_n)=(n-1)/(a_1a_n) .

"If "a_1,a_2,a_3,.....,a_n" are in AP, prove that "a_(1)+a_(n)=a_(r)+a_(n-r+1)""

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 1, a_1,a_2,a_3 ,…, a_(n-1) are the nth roots of unity then prove that : (1-a_1)(1-a_2)(1-a_3)...(1-a_(n-1)) =n.

If a_1,a_2,a_3,...,a_n are in AP and a_i ne (2k-1)pi/2 for all i , find the sum seca_1*seca_2+seca_2*seca_3+seca_3*seca_4+...+seca_(n-1)*seca_n .

RD SHARMA-ARITHMETIC PROGRESSIONS-Solved Examples And Exercises
  1. Show that in an A.P. the sum of the terms equidistant from the begi...

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

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

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

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

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

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

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  9. 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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  10. 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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  11. Find the sum of all those integers between 100 and 800 each of which o...

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

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

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

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  15. The digits of a positive integer, having three digits, are in A.P. and...

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  16. If x ,y ,z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M...

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  17. A man repays a loan of R s .3250 by paying R s .20 in the first month ...

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  18. If (b+c-a)/a ,(b+c-a)/b ,(a+b-c)/c , are in A.P., prove that 1/a ,1/b ...

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  19. If (b-c)^2,(c-a)^2,(a-b)^2 are in A.P., then prove that 1/(b-c),1/(c-a...

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  20. If a ,b ,c are in A.P., prove that 1/(b c),1/(c a),1/(a b), is al...

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