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For all n in N, 1 + 1/(sqrt(2))+1/(sqrt(...

For all `n in N, 1 + 1/(sqrt(2))+1/(sqrt(3))+1/(sqrt(4))++1/(sqrt(n))`

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Explore conceptually related problems

Statement-1: For every natural number nge2 , (1)/(sqrt(1))+(1)/(sqrt(2))+(1)/(sqrt(3))+...+(1)/(sqrt(n))gtsqrt(n) Statement-2: For every natural number nge2, sqrt(n(n+1))ltn+1

lim_ (n rarr oo) sum_ (n = 1) ^ (n) (sqrt (n)) / (sqrt (r) (3sqrt (r) + 4sqrt (n)) ^ (2))

lim_(n rarr oo)(sin(1)/(sqrt((n))))((1)/(sqrt(n+1)))^(+(1)/(sqrt(n+2))+(1)/(sqrt(n+2)))

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 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)))

Use induction to show that for all n in N . sqrt(a+sqrt(a+sqrt(a+....+sqrt(a))))lt (1+sqrt((4a+1)))/(2) where'a' is fixed positive number and n radical signs are taken on LHS.

MOTION-SEQUENCE & SERIES -Exercise -4 Level -II Previous Year /JEE Advanced
  1. For all n in N, 1 + 1/(sqrt(2))+1/(sqrt(3))+1/(sqrt(4))++1/(sqrt(n))

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  2. Let Vr denote the sum of first r terms of an arithmetic progression (A...

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

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

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

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

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

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  8. A straight line through the vertex P of a triangle PQR intersects the ...

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  9. Suppose four distinct positive numbers a1,a2,a3,a4 are in G.P. Let b1=...

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  10. In the sum of first n terms of an A.P. is cn^2, then the sum of square...

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  11. Let Sk ,k=1,2, ,100 , denotes thesum of the infinite geometric series ...

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  12. Let a(1),a(2),a(3),"........."a(11) be real numbers satisfying a(1)=15...

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  13. The minimum value of the sum of real number a^(-5),a^(-4),3a^(-3),1,a^...

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  14. Let a1, a2, a3, ,a(100) be an arithmetic progression with a1=3a n dsp...

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  15. If a(1),a(2),a(3),"......" be in harmonic progression with a(1)=5 and ...

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  16. Let a(n) denote the number of all n-digit numbers formed by the digits...

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  17. Let an denote the number of all n-digit positive integers formed by th...

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  18. Let S(n)=sum(k=1)^(4n)(-1)^((k(k+1))/2)k^(2). Then S(n) can take value...

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  19. A pack contains n cards numbered from 1 to n. Two consecutive numbered...

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  20. Let a,b,c be positive integers such that (b)/(a) is an integer. If a,b...

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  21. Suppose that all the terms of an arithmetic progression (A.P.) are nat...

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