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Find the sum of the products of the inte...

Find the sum of the products of the integers 1,2,3,....n taken two at a time and show that it equal to half the excess of the sum of the cubes of the given integers over the sum of their squares.

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`1/2[(1+2+3+....+n)^2-(1^2+2^2+3^2+....n^2)]`
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PATHFINDER-PROGRESSION AND SERIES -QUESTION BANK
  1. Prove that the number of the sequence 121,12321,1234321,...... are eac...

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  2. Find the sum of n terms of series 1+5((4n+1)/(4n-3))+9((4n+1)/(4n-3))^...

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  3. Find the sum of the products of the integers 1,2,3,....n taken two at ...

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  4. Sum the series n+(n-1)x+(n-2)x^2+.......+2x^(n-2)+x^(n-1).

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  5. Find 1+2^2x+3^2x^2+4^2x^3+.... to infty absx lt 1.

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  6. Find the sum of 1st n terms of the sequence 3,6,15,42,123,...

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  7. Let Sn denote the sum of first n terms of the sequence 1,5,14,30,55,.....

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  8. Finds the sum of first n terms of the series 3/(1^2xx2^2)+5/(2^2xx3^2)...

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  9. How many terms of the series 54+51+48+45+ ... must be taken to make 51...

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  10. If (b+c),(c+a),(a+b) are in H.P. show that a^2 ,b^2 , c^2 are in A.P.

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  11. Find the sum of first n terms of the series 3/1^2+5/(1^2+2^2)+7/(1^2+2...

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  12. If 0 < theta < pi/2 then find the least value of tantheta+cottheta

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  13. If x and y are positive quantities whose sum is 4, show that (x+1/x)^2...

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  14. If a,b,c gt 0 show that (bc)/(b+c)+(ca)/(c+a)+(ab)/(a+b)le(a+b+c)/2.

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  15. Show that a^2(1+b^2)+b^2(1+c^2)+c^2(1+a^2)ge6abc.

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  16. If m, n are positive quantities, prove that ((mn+1)/(m+1))^(m+1)gen^m.

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  17. Prove that ((bc+ac+ab)/(a+b+c))^(a+b+c)ge(b)^a(c)^b(a)^c [where a,b,c ...

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  18. If log2,log(2^n-1) and log(2^n+3) are in A.P. then n =

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  19. If the ratiio of the sum of n terms ofd two AP's is (3n+1):(2n+3) then...

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  20. If a1,a2,a3,.......an are in AP where a1 gt 0foralli then the value of...

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