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Sequences and Series

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  1. 101.Let the first term of a series be T_1=6 and its r^ term T_r=3T_r-1+6^r,r=2,3, ............
  2. 102.If 1+√3-√22√3+5-2√618+9√3-11√236√3+49-20√6180+ upto =2+((b)/(a)+1) _e((a)/(b)), where a an…
  3. 103.Let [a_1, a_2, a_3, ] be in an arithmetic progression of positive terms.
  4. 104.If three successive terms of a G.P.
  5. 105.Let 3,7,11,15, , 403 and 2,5,8,11, , 404 be two arithmetic progressions.
  6. 106.Let S_n be the sum to n-terms of an arithmetic progression 3,7,11, If 40<((6)/(n(n+1)) _k=…
  7. 107.Let =1^2+4^2+8^2+13^2+19^2+26^2+ upto 10 terms and = _n=1^10n^4.
  8. 108.If 8=3+(1)/(4)(3+p)+(1)/(4^2)(3+2 p)+(1)/(4^3)(3+3 p)+ , then the value of p is __________…
  9. 109.If the sum of the series ((1)/(2)-(1)/(3))+(12^2-(1)/(2 3)+13^2)+(12^3-12^2 3+12 3^2-13^3)…
  10. 110.The sum to 20 terms of the series [2 2^2-3^2+2 4^2-5^2+2 6^2- ] is equal to __________.
  11. 111.For k N, if the sum of the series 1+(4)/(k)+8k^2+13k^3+19k^4+ is 10 , then the value of k …
  12. 112.Let [S=109+(108)/(5)+1075^2+ .+25^107+15^108].
  13. 113.Suppose a_1, a_2, 2, a_3, a_4
  14. 114.The sum of all those terms, of the arithmetic progression 3, 8, 13, ...., 373, which are n…
  15. 115.Let 0 < z < y < x be three real numbers such that [(1)/(x), (1)/(y), (1)/(z)] are in an ar…
  16. 116.If [(20)^19+2(21)(20)^18+3(21)^2(20)^17++20(21)^19=k(20)^19], then k is equal to _________…
  17. 117.The sum 1^2-2 3^2+3 5^2-4 7^2+5 9^2-+15 29^2 is _________.
  18. 118.The 8^ common term of the series gathered S_1=3+7+11+15+19+ ..
  19. 119.Consider an arithmetic series and a geometric series having four initial terms from the se…
  20. 120.Let a_1 = 1 and, for n 1, [a_n+1 = (a_n)/(2) + (2)/((n+1)^2) - (1)/(n^2).] Then -_n=1^(a_n…
  21. 121.Let a, b, c be in A.P.
  22. 122.If _r=1^25 (r)/(r^4+r^2+1) = (p)/(q), where p, q are positive integers such that (p,q)=1, …
  23. 123.If _k=1^n a_k = 6n^3 for every n N, then _k=1^6(a_k+1-a_k36)^2 is equal to ________.
  24. 124.Let f() = ^2 + ^2 and g() = ^2 + ^2, where > > 0.
  25. 125.Let a_n be a G.P.
  26. 126.Let a_1, a_2, , a_10 be positive real numbers such that [(a_2)/(2a_1) = (a_3)/(2a_2) = = a…
  27. 127.Let _k=1^n a_k = n^2 + n for all n.
  28. 128.Consider an A.P.
  29. 129.Let 729, 81, 9, 1, be a G.P., and let P_n denote the product of the first n terms of this …
  30. 130.Let a_1, a_2, a_3, a_4 be an A.P.
  31. 131.The value of _k=1^ (-1)^k+1(k(k+1))/(k!) is equal to:
  32. 132.The sum [63^26 + 1013^25 + 1023^24 + 102^23^23 + + 102^243^3] is equal to:
  33. 133.Let the arithmetic mean of 1a and 1b be 516, with a>2.
  34. 134.Let A be the set of the first 101 terms of the A.P.
  35. 135.The sum T_1+T_2++T_8, where T_r = 1^3+2^3++r^31+3+5++(2r-1), is:
  36. 136._n=1^10 (528)/(n(n+1)(n+2)) is equal to:
  37. 137.Let the sum of the first n terms of an A.P.
  38. 138.Let A_1, A_2, , A_39 be 39 arithmetic means between the numbers 59 and 159.
  39. 139.If the sum of the first 10 terms of the series [(1)/(1+1^4) + (2)/(1+2^4) + (3)/(1+3^4) + …
  40. 140.If T_n = 1n(1^2+2^2++n^2), the sum T_1+T_2++T_10 (up to 10 terms) is equal to:
  41. 141.Consider the quadratic equation (n^2-2n+2)x^2-3x+(n^2-2n+2)^2=0, nR.
  42. 142.The roots of the quadratic equation 3x^2 - px + q = 0 are the 10th and 11th terms of an ar…
  43. 143.Let a_1,a_2,,a_2024 be an Arithmetic Progression such that a_1+(a_5+a_10+a_15++a_2020)+a_2…
  44. 144.If the sum of the first 10 terms of the series [(41)/(1+41^4) + (42)/(1+42^4) + (43)/(1+43…
  45. 145.Let a_1,a_2,a_3, be a G.P.
  46. 146.Suppose that the number of terms in an A.P.
  47. 147.Let S_n = 12+16+112+120+ up to n terms.
  48. 148.Let T_r be the r-th term of an A.P.
  49. 149.Let a_n be a sequence such that a_0=0, a_1=12, and 2a_n+2=5a_n+1-3a_n for n=0,1,2,.
  50. 150.For positive integers n, if 4a_n = n^2+5n+6 and S_n = _k=1^n(1)/(a_k), then the value of 5…