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Sequences and Series

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  1. 251.Let S_n denote the sum of the first n terms of an arithmetic progression.
  2. 252.The sum of all the elements in the set n1,2,,100 : (n,2040)=1 is equal to ________.
  3. 253.If the sum of the first 21 terms of the series _9^1/2x+_9^1/3x+_9^1/4x+, where x>0, is 504…
  4. 254.For > 0, let 1(+1)(+2)(+20) = _k=0^20(A_k)/(+k).
  5. 255.If the value of 1+23+63^2+103^3+ is , and _0.25(13+13^2+13^3+) is , and L=^, then L^2 is e…
  6. 256.Let a_n_n=1^ be a sequence such that a_1=1, a_2=1, and a_n+2=2a_n+1+a_n for all n1.
  7. 257.Let a_1,a_2,a_3, be an A.P.
  8. 258.The number of 4-digit numbers which are neither a multiple of 7 nor a multiple of 3 is ___…
  9. 259.Three numbers are in an increasing geometric progression with common ratio r.
  10. 260.If 0<x<1, then 32x^2+53x^3+74x^4+ is equal to:
  11. 261.If x,yR, x>0, y = _10x+_10x^1/3+_10x^1/9+ up to terms, and 2+4+6++2y3+6+9++3y = 4_10x, the…
  12. 262.If 0<x<1 and y = 12x^2+23x^3+34x^4+, then the value of e^1+y at x=12 is:
  13. 263.If the sum of an infinite G.P.
  14. 264.Let a_1,,a_10 be an A.P.
  15. 265.The sum of 10 terms of the series 31^22^2+52^23^2+73^24^2+ is:
  16. 266.If S = 75+95^2+135^3+195^4+, then 160S is equal to ________.
  17. 267.Let a_1,a_2,,a_21 be an A.P.
  18. 268.Let S_n = 1(n-1)+2(n-2)+3(n-3)++(n-1)1, n4.
  19. 269.The number of terms common to the two A.P.'s 3,7,11,,407 and 2,9,16,,709 is ________.
  20. 270.If the 10th term of an A.P.
  21. 271.Let f:RR be such that for all xR, (2^1+x+2^1-x), f(x), (3^x+3^-x) are in A.P.
  22. 272.Five numbers are in A.P., whose sum is 25 and product is 2520.
  23. 273.The product 2^1/44^1/168^1/4816^1/128 to is equal to:
  24. 274.Let a_n be the nth term of a G.P.
  25. 275.If x = _n=0^(-1)^n^2n and y=_n=0^^2n, for 0<<4, then:
  26. 276.The greatest positive integer k, for which 49^k+1 is a factor of the sum 49^125+49^124++49…
  27. 277.Let a_1,a_2,a_3, be a G.P.
  28. 278.The coefficient of x^7 in the expression (1+x)^10+x(1+x)^9+x^2(1+x)^8++x^10 is:
  29. 279.The sum _n=1^7(n(n+1)(2n+1))/(4) is equal to ________.
  30. 280.The sum _k=1^20(1+2+3++k) is ________.
  31. 281.If the sum of the first 40 terms of the series 3+4+8+9+13+14+18+19+ is (102)m, then m is e…
  32. 282.If the sum of the first 11 terms of an A.P., a_1,a_2,a_3, is 0 (a_10), then the sum of the…
  33. 283.If the first term of an A.P.
  34. 284.If the sum of the series 20+1935+1915+1845+ up to the nth term is 488, and the nth term is…
  35. 285.Let a_1,a_2,,a_n be a given A.P.
  36. 286.If 3^2^2-1, 14, 3^4-2^2 are the first three terms of an A.P.
  37. 287.If the sum of the first 20 terms of the series _7^1/2x+_7^1/3x+_7^1/4x+ is 460, then x is …
  38. 288.If f(x+y)=f(x)f(y) and _x=1^f(x)=2, x,yN, then the value of f(4)f(2) is:
  39. 289.Let a,b,c,d,p be any nonzero distinct real numbers such that (a^2+b^2+c^2)p^2 - 2(ab+bc+cd…
  40. 290.The common difference of the A.P.
  41. 291.Suppose that a function f:RR satisfies f(x+y)=f(x)f(y) for all x,yR, and f(1)=3.
  42. 292.If 2^10+2^93^1+2^83^2++23^9+3^10 = S-2^11, then S is equal to:
  43. 293.If the sum of the second, third and fourth terms of a positive-term G.P.
  44. 294.Let , be the roots of x^2-3x+p=0 and , be the roots of x^2-6x+q=0.
  45. 295.The value of (0.16)^_2.5(13+(1)/(3^2)+(1)/(3^3)+) is equal to ________.
  46. 296.The sum of the first three terms of a G.P.
  47. 297.If |x|<1, |y|<1, and x y, then the sum to infinity of the series (x+y)+(x^2+xy+y^2)+(x^3+x…
  48. 298.Let S be the sum of the first 9 terms of the series x+ka+x^2+(k+2)a+x^3+(k+4)a+x^4+(k+6)a+…
  49. 299.If m arithmetic means (A.M.s) and three geometric means (G.M.s) are inserted between 3 and…
  50. 300.If 1+(1-2^21)+(1-4^23)+(1-6^25)++(1-20^219) = -220, then the ordered pair (,) is equal to: