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Sequences and Series

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  1. 51.In an increasing geometric progression of positive terms, the sum of the second and sixth …
  2. 52.Let A B C be an equilateral triangle.
  3. 53.A software company sets up m number of computer systems to finish an assignment in 17 days…
  4. 54.For x 0, the least value of K, for which 4^1+x+4^1-x,(K)/(2),16^x+16^-x are three consecut…
  5. 55.If 1√1+√2+1√2+√3+ +1√99+√100=m and (1)/(1 2)+(1)/(2 3)+ +(1)/(99 100)=n, then the point (m…
  6. 56.The value of [(1 2^2+2 3^2++100 (101)^2)/(1^2 2+2^2 3+ .+100^2 101)] is
  7. 57.Let three real numbers a, b, c be in arithmetic progression and a+1, b, c+3 be in geometri…
  8. 58.Let the first three terms 2, p and q, with q 2, of a G.P.
  9. 59.Let S_n denote the sum of the first n terms of an arithmetic progression.
  10. 60.Let 3, a, b, c be in A.P.
  11. 61.Let 2^,8^ and 44^ terms of a non-constant A.
  12. 62.For 0(I) If (-1,0), then b cannot be the geometric mean of a and c (II) If (0,1), then b m…
  13. 63.The sum of the series [(1)/(1-3 1^2+1^4)+(2)/(1-3 2^2+2^4)+(3)/(1-3 3^2+3^4)+] up to 10 -t…
  14. 64.Let a and b be be two distinct positive real numbers.
  15. 65.Let S_n denote the sum of first n terms of an arithmetic progression.
  16. 66.If _ea, _eb, _ec are in an A.P.
  17. 67.If each term of a geometric progression [a_1, a_2, a_3, ] with [a_1=(1)/(8)] and [a_2 a_1]…
  18. 68.If in a G.P.
  19. 69.In an A.P., the sixth term a_6=2.
  20. 70.20^20,19(1)/(4),18(1)/(2),17(3)/(4), ,-129(1)/(4)
  21. 71.The number of common terms in the progressions 4,9,14,19, , up to 25^ th term and 3,6,9,12…
  22. 72.Let A_1 and A_2 be two arithmetic means and G_1, G_2, G_3 be three geometric means of two …
  23. 73.Let a_1, a_2, a_3, ....
  24. 74.Let [s_1, s_2, s_3, , s_10] respectively be the sum to 12 terms of 10 A.P.
  25. 75.Let a, b, c and d be positive real numbers such that a+b+c+d=11.
  26. 76.Let [x_1, x_2, , x_100] be in an arithmetic progression, with x_1=2 and their mean equal t…
  27. 77.If S_n=4+11+21+34+50+ to n terms, then (1)/(60)(S_29-S_9) is equal to :
  28. 78.Let the first term and the common ratio r of a geometric progression be positive integers.
  29. 79.Let a_n be the n^ term of the series 5+8+14+23+35+50+ and S_n= _k=1^na_k.
  30. 80.Let S_K=(1+2+ +K)/(K) and _j=1^nS_2^j=(n)/(A)(Bn^2+Cn+D), where A,B,C,D N and A has least …
  31. 81.If gcd(m,n)=1 and 1^2-2^2+3^2-4^2+ ..+(2021)^2-(2022)^2+(2023)^2=1012m^2n then m^2-n^2 is …
  32. 82.The sum of the first 20 terms of the series [5+11+19+29+41+] is :
  33. 83.Let a_1, a_2, a_3, be an A.P.
  34. 84.Let a, b, c>1, a^3, b^3 and c^3 be in A.P., and _a b, _c a and _b c be in G.P.
  35. 85.The minimum value of 2sinx + 2cosx is :
  36. 86.For the functions f( )= ^2 + ^2 , and g( )= ^2 + ^2 , > >0, let 0< <( )/(2)minf( )=0< < ma…
  37. 87.If _k=1^na_k=6n^3, then _k=1^6(a_k+1-a_k36)^2 is equal to ________.
  38. 88.If _r=1^25(rr^4+r^2+1)=(p)/(q), where p and q are positive integers such that gcd(p,q)=1, …
  39. 89.In a G.P., if the product of the first three terms is 27 and the set of all possible value…
  40. 90.Suppose a,b,c are in A.P.
  41. 91.Let a_1=1 and for n 1,a_n+1=(1)/(2)a_n+n^2-2n-1n^2(n+1)^2.
  42. 92.If the sum of the first 10 terms of the series (4 1)/(1+4 1^4)+(4 2)/(1+4 2^4)+(4 3)/(1+4 …
  43. 93.Let a_1,a_2, ,a_2024 be an Arithmetic Progression such that a_1+(a_5+a_10+a_15+ +a_2020)+a…
  44. 94.The interior angles of a polygon with n sides, are in an A.P.
  45. 95.The roots of the quadratic equation 3x^2-px+q=0 are 10^ and 11^ terms of an arithmetic pro…
  46. 96.If ((1)/( +1)+(1)/( +2)+ ..+(1)/( +1012))-((1)/(2 1)+(1)/(4 3)+(1)/(6 5)+ +(1)/(2024 2023)…
  47. 97.An arithmetic progression is written in the following way The sum of all the terms of the …
  48. 98.Let the positive integers be written in the form : If the k^ row contains exactly k number…
  49. 99.Let = _r=0^n(4r^2+2r+1)^nC_r and =( _r=0^n^nC_rr+1)+(1)/(n+1).
  50. 100.If S(x)=(1+x)+2(1+x)^2+3(1+x)^3+ +60(1+x)^60,x 0, and (60)^2S(60)=a(b)^b+b, where a,b N, t…