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S=sum(i=1)^nsum(j=1)^isum(k=1)^j1...

`S=sum_(i=1)^nsum_(j=1)^isum_(k=1)^j1`

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`sum_(i=1)^(n)sum_(j=1)^(i)sum_(k=1)^(j)1=sum_(i=1)^(n)(i(i+1))/(2)`
`=(1)/(2)[sum_(i=1)^(n)i^(2)+sum_(i=1)^(i)i]=(1)/(2)[sumn^(2)+sumn]`
`=(1)/(2)[(n(n+1)(2n+1))/(6)+(n(n+1))/(2)]`
`=(n(n+1))/(12)[2n+1+3]=(n(n+1)(n+2))/(6)`
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ARIHANT MATHS-SEQUENCES AND SERIES-Exercise (Questions Asked In Previous 13 Years Exam)
  1. S=sum(i=1)^nsum(j=1)^isum(k=1)^j1

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

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  3. If an=3/4-(3/4)^2+(3/4)^3+...(-1)^(n-1)(3/4)^n and bn=1-an, then find ...

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  4. If a1, a2, a3, be terms of an A.P. if (a1+a2++ap)/(a1+a2++aq)=(p^2)/(...

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

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

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

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

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

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  12. in a geometric progression consisting of positive terms, each term eq...

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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 su...

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  15. If the sum of first n terms of an AP is cn^(2), then the sum of square...

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  16. 1+2/3+6/(3^2)+10/(3^3)+14/(3^4)+...oo=

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

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

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  19. A person is to cout 4500 currency notes. Let a(n) denotes the number o...

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

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  21. A man saves Rs. 200 in each of the first three months of his service. ...

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