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Sequences and Series

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  1. 201.The sum 1+23+33^2++103^9 is equal to:
  2. 202.Let A = _i=1^10_j=1^10i,j and B = _i=1^10_j=1^10i,j.
  3. 203.If A = _n=1^(1)/((3+(-1)^n)^n) and B = _n=1^((-1)^n)/((3+(-1)^n)^n), then AB is equal to:
  4. 204.If a_1 (>0), a_2, a_3, a_4, a_5 are in a G.P., a_2+a_4 = 2a_3+1, and 3a_2+a_3 = 2a_4, then…
  5. 205.x=_n=0^a^n, y=_n=0^b^n, z=_n=0^c^n, where a,b,c are in A.P.
  6. 206.If the sum of the first ten terms of the series 15+265+3325+41025+52501+ is mn, where m,n …
  7. 207.Let S = 2+67+127^2+207^3+307^4+.
  8. 208.If a_1,a_2,a_3, and b_1,b_2,b_3, are A.P.s, and a_1=2, a_10=3, a_1b_1=1=a_10b_10, then a_4…
  9. 209.Let A_1, A_2, A_3, be an increasing geometric progression of positive real numbers.
  10. 210.If n arithmetic means are inserted between a and 100 such that the ratio of the first mean…
  11. 211.For n=1,2,,50, let S_n be the sum of the infinite geometric progression whose first term i…
  12. 212.Let a_n_n=0^ be a sequence such that a_0=a_1=0 and a_n+2 = 2a_n+1-a_n+1 for all n0.
  13. 213.The sum of the infinite series 1+56+126^2+226^3+356^4+516^5+706^6+ is equal to:
  14. 214.Let 3,6,9,12, up to 78 terms and 5,9,13,17, up to 59 terms be two series.
  15. 215.Let a_1=b_1=1, a_n = a_n-1+2, and b_n = a_n+b_n-1 for every natural number n2.
  16. 216.The sum _n=1^21(3)/((4n-1)(4n+3)) is equal to:
  17. 217.Consider two G.P.s 2,2^2,2^3, (60 terms) and 4,4^2,4^3, (n terms).
  18. 218.The series of positive multiples of 3 is divided into sets: 3, 6,9,12, 15,18,21,24,27,.
  19. 219.If _k=1^10(k)/(k^4+k^2+1) = (m)/(n), where m,n are coprime, then m+n is equal to ________.
  20. 220.Different A.P.'s are constructed with first term 100, last term 199, and integral common d…
  21. 221.Suppose a_1,a_2,,a_n, is an arithmetic progression of natural numbers.
  22. 222.Let f(x)=2x^2-x-1 and S=nZ: |f(n)|800.
  23. 223.Let the sum of an infinite G.P., whose first term is a and common ratio is r, be 5.
  24. 224.[(2^3-1^3)/(17)+(4^3-3^3+2^3-1^3)/(211)+(6^3-5^3+4^3-3^3+2^3-1^3)/(315)++(30^3-29^3++2^3-1…
  25. 225.Consider the sequence a_1,a_2,a_3, such that a_1=1, a_2=2, and a_n+2=2a_n+1+a_n for n=1,2,…
  26. 226.For p,qR, consider the real-valued function f(x)=(x-p)^2-q, xR, q>0.
  27. 227.Let x_1,x_2,,x_20 be in geometric progression with x_1=3 and common ratio 12.
  28. 228.[13^12 + 5(2^03^12+2^13^11+2^23^10++2^113) = 2^n m,] where m is odd.
  29. 229.If 1(20-a)(40-a)+1(40-a)(60-a)++1(180-a)(200-a)=1256, then the maximum value of a is:
  30. 230.Let a_1,a_2,a_3, be an A.P.
  31. 231.If 1234+1345+1456++1100101102=k101, then 34k is equal to ________.
  32. 232.Let a_n_n=0^ be a sequence such that a_0=a_1=0 and a_n+2=3a_n+1-2a_n+1 for all n0.
  33. 233._r=1^20(r^2+1)(r!) is equal to:
  34. 234.If e^(^2x+^4x+^6x+)2 satisfies the equation t^2-9t+8=0, then the value of 2 x x+√3 x, 0<x<…
  35. 235.Let A = x: x is a 3-digit number, B=x: x=9k+2, kZ, and C=x: x=9k+, kZ, 0<<9 for some .
  36. 236.The sum of the first four terms of a geometric progression (G.P.) is 6512, and the sum of …
  37. 237.The sum of the series _n=1^(n^2+6n+10)/((2n+1)!) is equal to:
  38. 238.The sum of the infinite series 1+23+73^2+123^3+173^4+223^5+ is equal to:
  39. 239.If the arithmetic mean and geometric mean of the pth and qth terms of the sequence -16, 8,…
  40. 240.In an increasing geometric series, the sum of the second and sixth term is 252, and the pr…
  41. 241.The minimum value of f(x) = a^a^x+a^1-a^x, where a,xR and a>0, is equal to:
  42. 242.If 0<,<2, x=_n=0^^2n, y=_n=0^^2n, and z=_n=0^^2n^2n, then:
  43. 243.Let A_1, A_2, A_3, be squares such that, for each n1, the length of the side of A_n equals…
  44. 244.Let 116, a, b be in G.P.
  45. 245.If , are natural numbers such that 100-199 = (100)(100)+(99)(101)+(98)(102)++(1)(199), the…
  46. 246.13^2-1+15^2-1+17^2-1++1201^2-1 is equal to:
  47. 247.Let S_1 be the sum of the first 2n terms of an arithmetic progression, and S_2 the sum of …
  48. 248.If _32, _3(2^x-5), _3(2^x-72) are in an arithmetic progression, then the value of x is equ…
  49. 249.If 9, x, 718 are in arithmetic progression, and 9, y, 518 are also in arithmetic progressi…
  50. 250.Let S_n be the sum of the first n terms of an arithmetic progression.