Sequences and Series

For x \(\varepsilon\) R, let [x] denote the greatest integer \(\le\) x, then the sum of the series
$$\left[ { - {1 \over 3}} \right] + \left[ { - {1 \over 3} - {1 \over {100}}} \right] + \left[ { - {1 \over 3} - {2 \over {100}}} \right] ...

Let Sn denote the sum of the first n terms of an A.P. If S4 = 16 and S6= – 48, then S10 is equal to :

If a1, a2, a3, ..... are in A.P. such that a1 + a7 + a16 = 40, then the sum of the first 15 terms of this A.P. is :

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is \({{27} \over {19}}\).Then the common ratio of this series is :

Let a1, a2, . . . . . ., a10 be a G.P.    If \({{{a_3}} \over {{a_1}}} = 25,\) then \({{{a_9}} \over {{a_5}}}\) equals

If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is :

Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression \({{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}}\) is :

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is -

Let a1, a2, a3, ..... a10 be in G.P. with ai > 0 for i = 1, 2, ….., 10 and S be the set of pairs (r, k), r, k \(\in\) N (the set of natural numbers) for which
$$\left| {\matrix{
{{{\log }_e}\,{a_1}^r{a_2}^k} & {{{\log }_e}\,{a_2}^r{a_3...

If a1, a2, a3, ............... an are in A.P. and a1 + a4 + a7 + ........... + a16 = 114, then a1 + a6 + a11 + a16 is equal to :

The sum
\({{3 \times {1^3}} \over {{1^3}}} + {{5 \times ({1^3} + {2^3})} \over {{1^2} + {2^2}}} + {{7 \times \left( {{1^3} + {2^3} + {3^3}} \right)} \over {{1^2} + {2^2} + {3^2}}} + .....\) upto 10 terms is:

Let a1, a2, a3,......be an A.P. with a6 = 2. Then the common difference of this A.P., which maximises the
product a1a4a5, is :

Let \(a\), b and c be in G.P. with common ratio r, where \(a\) \(\ne\) 0 and 0 < r \(\le\) \({1 \over 2}\)
. If 3\(a\), 7b and 15c are the first three
terms of an A.P., then the 4th term of this A.P. is :

The sum
\(1 + {{{1^3} + {2^3}} \over {1 + 2}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 2 + 3}} + ...... + {{{1^3} + {2^3} + {3^3} + ... + {{15}^3}} \over {1 + 2 + 3 + ... + 15}}\)\(- {1 \over 2}\left( {1 + 2 + 3 + ... + 15} \right)\) is equal...

The sum of the first 20 terms of the series
\(1 + {3 \over 2} + {7 \over 4} + {{15} \over 8} + {{31} \over {16}} + ...,\) is :

Let \({1 \over {{x_1}}},{1 \over {{x_2}}},...,{1 \over {{x_n}}}\,\,\) (xi \(\ne\) 0 for i = 1, 2, ..., n) be in A.P. such that x1=4 and x21 = 20. If n is the least positive integer for which \({x_n} > 50,\) then $$\sum\limits_{i = 1}^n {\...

If x1, x2, . . ., xn and \({1 \over {{h_1}}}\), \({1 \over {{h_2}}}\), . . . , \({1 \over {{h_n}}}\) are two A.P..s such that x3 = h2 = 8 and x8 = h7 = 20, then x5.h10 equals :

If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :

Let    An = \(\left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3}\) \(-\). . . . . + (\(-\)1)n-1 \({\left( {{3 \over 4}} \right)^n}\)    and    Bn = 1 \(-\) An.
Then, the least dd natural numb...

If  a,   b,   c  are in A.P. and  a2,  b2,  c2 are in G.P. such that
a < b < c and   a + b + c = \({3 \over 4},\) then the value of a is :

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series
12 + 2.22 + 32 + 2.42 + 52 + 2.62 ...........
If B - 2A = 100\(\lambda\), then \(\lambda\) is equal to

Let \({a_1}\), \({a_2}\), \({a_3}\), ......... ,\({a_{49}}\) be in A.P. such that
\(\sum\limits_{k = 0}^{12} {{a_{4k + 1}}} = 416\) and \({a_9} + {a_{43}} = 66\).
\(a_1^2 + a_2^2 + ....... + a_{17}^2 = 140m\), then m is equal to

Let
Sn = \({1 \over {{1^3}}}\)\(+ {{1 + 2} \over {{1^3} + {2^3}}} + {{1 + 2 + 3} \over {{1^3} + {2^3} + {3^3}}} + ......... + {{1 + 2 + ....... + n} \over {{1^3} + {2^3} + ...... + {n^3}}}.\)
If 100 Sn = n, then n is equal to :

If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :

If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then \({{a + b} \over {a - b}}\) is equal to :

If the sum of the first n terms of the series \(\,\sqrt 3 + \sqrt {75} + \sqrt {243} + \sqrt {507} + ......\) is \(435\sqrt 3 ,\) then n equals :

For any three positive real numbers a, b and c,
9(25\({a^2}\) + b2) + 25(c2 - 3\(a\)c) = 15b(3\(a\) + c).
Then

Let x, y, z be positive real numbers such that x + y + z = 12 and x3y4z5 = (0.1) (600)3. Then x3 + y3 + z3is equal to :

Let a1, a2, a3, . . . . . . . , an, . . . . . be in A.P.
If a3 + a7 + a11 + a15 = 72,
then the sum of its first 17 terms is equal to :

If   A > 0, B > 0   and    A + B = \({\pi \over 6}\), then the minimum value of tanA + tanB is :

Let z = 1 + ai be a complex number, a > 0, such that z3 is a real number.
Then the sum 1 + z + z2 + . . . . .+ z11 is equal to :

If the \({2^{nd}},{5^{th}}\,and\,{9^{th}}\) terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

If the sum of the first ten terms of the series \({\left( {1{3 \over 5}} \right)^2} + {\left( {2{2 \over 5}} \right)^2} + {\left( {3{1 \over 5}} \right)^2} + {4^2} + {\left( {4{4 \over 5}} \right)^2} + .......is\,{{16} \over 5}m,\) then m i...

The sum of first 9 terms of the series.
\({{{1^3}} \over 1} + {{{1^3} + {2^3}} \over {1 + 3}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 3 + 5}} + ......\)

If m is the A.M. of two distinct real numbers l and n \((l,n > 1)\) and \({G_1},{G_2}\) and \({G_3}\) are three geometric means between \(l\) and n, then \(G_1^4\, + 2G_2^4\, + G_3^4\) equals:

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :

If \({(10)^9} + 2{(11)^1}\,({10^8}) + 3{(11)^2}\,{(10)^7} + ......... + 10{(11)^9} = k{(10)^9},\), then k is equal to :

The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,........,is

Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000.
Statement-2: \(\sum\limits_{k = 1}^n {\left( {{k^3} - {{(k - 1)}^3}} \right)} = {n^3}\), for any natural number n.

A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 110...

A person is to count 4500 currency notes. Let \({a_n}\) denote the number of notes he counts in the \({n^{th}}\) minute. If \({a_1}\) = \({a_2}\) = ....= \({a_{10}}\)= 150 and \({a_{10}}\), \({a_{11}}\),.... are in an AP with common differe...

The sum to infinite term of the series \(1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + .....\) is

The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is equals

The sum of series \({1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}} - .......\) upto infinity is

Let \({a_1}\), \({a_2}\), \({a_3}\).....be terms on A.P. If \({{{a_1} + {a_2} + .....{a_p}} \over {{a_1} + {a_2} + .....{a_q}}} = {{{p^2}} \over {{q^2}}},\,p \ne q,\,then\,{{{a_6}} \over {{a_{21}}}}\,\) equals

If \({{a_1},{a_2},....{a_n}}\) are in H.P., then the expression \({{a_1}\,{a_2} + \,{a_2}\,{a_3}\, + .... + {a_{n - 1}}\,{a_n}}\) is equal to

If \(x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\infty {{c^n},} } } \,\,\) where a, b, c are in A.P and $$\,\left| a \right| < 1,\,\left| b \right| < 1,\,\left| c \right| <...

The sum of the series \(1 + {1 \over {4.2!}} + {1 \over {16.4!}} + {1 \over {64.6!}} + .......\) ad inf. is

Let \({{T_r}}\) be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, \(m \ne n,\,\,{T_m} = {1 \over n}\,\,and\,{T_n} = {1 \over m},\,\) then a - d equals