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If `a_(1),a_(2),a_(3),".....",a_(n)` are in HP, than prove that `a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)`

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To prove that if \( a_1, a_2, a_3, \ldots, a_n \) are in Harmonic Progression (HP), then \[ a_1 a_2 + a_2 a_3 + a_3 a_4 + \ldots + a_{n-1} a_n = (n-1) a_1 a_n, \] we will use the definition of HP and some algebraic manipulations. ...
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ARIHANT MATHS ENGLISH-SEQUENCES AND SERIES-Exercise (Questions Asked In Previous 13 Years Exam)
  1. If a(1),a(2),a(3),".....",a(n) are in HP, than prove that a(1)a(2)+a(2...

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  2. Let a,b,c be in A.P. and |a|lt1,|b|lt1|c|lt1.ifx=1+a+a^(2)+ . . . ."to...

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  3. about to only mathematics

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  4. If a1, a2, a3, be terms of an A.P. and (a1+a2+.....+ap)/(a1+a2+.....+...

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  5. If a1, a2, a3,.....an are in H.P. and a1 a2+a2 a3+a3 a4+.......a(n-1...

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  6. Let V(r ) denotes the sum of the first r terms of an arithmetic progre...

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  7. Let Vr denote the sum of the first r terms of an arithmetic progressio...

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  8. Let V(r) denote the sum of the first r terms of an arithmetic progress...

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  9. LetA(1),G(1),H(1) denote the arithmetic, geometric and harmonic means ...

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  10. Let A1 , G1, H1denote the arithmetic, geometric and harmonic means re...

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  11. LetA(1),G(1),H(1) denote the arithmetic, geometric and harmonic means ...

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  12. In a G.P of positive terms if any term is equal to the sum of the next...

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  13. Suppose four distinct positive numbers a(1),a(2),a(3),a(4) are in G.P....

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  14. The first two terms of a geometric progression add up to 12. The sum o...

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  15. If the sum of first n terms of an A.P. is cn^(2) then the sum of squar...

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  16. The sum to infinity of the series 1+2/3+6/3^2+14/3^4+...is

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  17. Let Sk,k=1, 2, …. 100 denote the sum of the infinite geometric series ...

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  18. Let a1, a2, a3, ,a(11) be real numbers satisfying a1=15 , 27-2a2>0 a ...

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  19. A person is to count 4500 currency notes. Let an denote the number of ...

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  20. The minimum value of the sum of real numbers a^-5, a^-4, 3a^-3, 1,a^8 ...

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  21. A man saves ₹ 200 in each of the first three months of his servies.In ...

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