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If the sequence a(1),a(2),a(3),…,a(n) is...

If the sequence `a_(1),a_(2),a_(3),…,a_(n)` is an A.P., then prove that `a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+…+a_(2n-1)^(2)-a_(2n)^(2)=n/(2n-1)(a_(1)^(2)-a_(2n)^(2))`

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A DAS GUPTA-Progression, Related Inequalities and Series-Exercise
  1. If a1,a2,a3,………an are in A.P, where ai.0 for all i show that 1/(sqrt(a...

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  2. Let a1,a2,a3 …. an be in A.P. If 1/(a1an)+1/(a2a(n-1))+… + 1/(an a...

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  3. If the sequence a(1),a(2),a(3),…,a(n) is an A.P., then prove that a(1)...

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  4. If the nonzero numbers a1,a2,a3,....,an are in AP, prove that 1/(a1a2a...

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  5. If a,b,c are in A.P. prove that b+c,c+a,a+b are also in A.P.

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  6. If ab+bc+ca!=0 and a,b,c are in A.P. prove that a^2(b+c), b^2(c+a), c^...

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  7. If a^2,b^2,c^2 are in A.P. prove that 1/(b+c),1/(c+a),1/(a+b) are in A...

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  8. If a^2, b^2, c^2 are in A.P, show that: a/(b+c), b/(c+a), c/(a+b) are...

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  9. If x, y, z are in AP, prove that 1/(sqrty+sqrtz),1/(sqrtz+sqrtx),1/(sq...

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  10. If Sn=an^2+bn+c where c ne 0 and Sn denotes the sum to n terms of a se...

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  11. The sum of n terms of a series is 2n^2+3n. Is the series arithmetic ? ...

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  12. Find the sum of all positive integers less than 200 that are divisible...

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  13. Find the sum of all natural numbers n in [100, 300] that are neither d...

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  14. Find the sum of all 3-digit natural numbers which are of the form 3m+2...

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  15. Sum the series to n terms: 1/(1+sqrtx)+1/(1-x)+1/(1-sqrtx)+.....

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  16. How many terms of 1+3+5+7+... amount to 1234321 ?

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  17. The maximum sum of the series 20+19 1/3+18 2/3+ is 310 b. 300 c. 0320 ...

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  18. If a1,a2...................,a19 are the first 19 term of an AP and a1+...

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  19. The sum of the first 100 terms common to the series 17, 21, 25, 29...

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  20. in an AP , Sp=q , Sq=p and Sr denotes the sum of the first r terms. Th...

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