Sequences and Series

Let $7 \overbrace{5 \cdots 5}^r 7$ denote the $(r+2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5 . Consider the sum $S=77+757+7557+\cdots+7 \overbrace{5 \cdots 5}^{98}7$. If $S=\frac{7 \overbra...

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be an arithmetic progression with \(a_{1}=7\) and common difference 8. Let \(T_{1}, T_{2}, T_{3}, \ldots\) be such that \(T_{1}=3\) and \(T_{n+1}-T_{n}=a_{n}\) for \(n \geq 1\). Then, which of the followi...

Let \(l_{1}, l_{2}, \ldots, l_{100}\) be consecutive terms of an arithmetic progression with common difference \(d_{1}\), and let \(w_{1}, w_{2}, \ldots, w_{100}\) be consecutive terms of another arithmetic progression with common differenc...

For any positive integer n, let Sn : (0, \(\infty\)) \(\to\) R be defined by \({S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)}\), where for any x \(\in\) R, $${\cot ^{ - 1}}(x) \in (0,\pi...

Let m be the minimum possible value of \({\log _3}({3^{{y_1}}} + {3^{{y_2}}} + {3^{{y_3}}})\), where \({y_1},{y_2},{y_3}\) are real numbers for which \({{y_1} + {y_2} + {y_3}}\) = 9. Let M be the maximum possible value of $$({\log _3}{x_1} ...

Let a1, a2, a3, .... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b1, b2, b3, .... be a sequence of positive integers in geometric progression with common ratio 2. If a1 = b1 = c, then the...

Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If \(AP(1;3) \cap AP(2;5) \cap AP(3;7)\) = AP(a ; d), then a + d equals ..............

Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, ...., and Y be the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, .... . Then, the number of elements in the set X ...

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

Let bi > 1 for I = 1, 2, ......, 101. Suppose logeb1, logeb2, ......., logeb101 are in Arithmetic Progression (A.P.) with the common difference loge2. Suppose a1, a2, ......, a101 are in A.P. such that a1 = b1 and a51 = b51. If t = b1 + b2 ...

The coefficient of \({x^9}\) in the expansion of (1 + x) (1 + \({x^2)}\) (1 + \({x^3}\)) ....\((1 + {x^{100}})\) is

Suppose that all the terms of an arithmetic progression (A.P) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the co...

Let a, b, c be positive integers such that \({b \over a}\) is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of \({{{a^2} + a - 14} \over {a + 1}}\) is

A pack contains \(n\) cards numbered from \(1\) to \(n.\) Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is \(1224.\) If the smaller of the numbers on the removed cards is \(k,\) t...

Let \({S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.\) Then \({S_n}\)can take value(s)

Let \({a_1},{a_2},{a_3},.....\) be in harmonic progression with \({a_1} = 5\) and \({a_{20}} = 25.\) The least positive integer \(n\) for which \({a_n} < 0\) is

Let \({{a_1}}\), \({{a_2}}\), \({{a_3}}\)........ \({{a_{100}}}\) be an arithmetic progression with \({{a_1}}\) = 3 and \({S_p} = \sum\limits_{i = 1}^p {{a_i},1 \le } \,p\, \le 100\). For any integer n with \(1\,\, \le \,n\, \le 20\), let m...

Let \({a_1},\,{a_{2\,}},\,{a_3}\)......,\({a_{11}}\) be real numbers satisfying \({a_1} = 15,27 - 2{a_2} > 0\,\,and\,\,{a_k} = 2{a_{k - 1}} - {a_{k - 2}}\,\,for\,k = 3,4,........11\). if $$\,\,\,{{a_1^2 + a_2^2 + .... + a_{11}^2} \over {11}...

Let \({S_k}\)= 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is \(\,{{k - 1} \over {k\,!}}\) and the common ratio is \({1 \over k}\). Then the value of $${{{{100}^2}} \over {100!}}\,\, + \,\,\sum\limits_{...

If the sum of first \(n\) terms of an A.P. is \(c{n^2}\), then the sum of squares of these \(n\) terms is

Suppose four distinct positive numbers \({a_1},\,{a_{2\,}},\,{a_3},\,{a_4}\,\) are in G.P. Let \({b_1} = {a_1},{b_2} = {b_1} + {a_2},\,{b_3} = {b_2} + {a_{3\,\,}}\,\,\,and\,\,\,{b_4} = {b_3} + {a_4}\).
STATEMENT-1: The numbers $${b_1},\,...

Let \({S_n} = \sum\limits_{k = 1}^n {{n \over {{n^2} + kn + {k^2}}}}\) and \({T_n} = \sum\limits_{k = 0}^{n - 1} {{n \over {{n^2} + kn + {k^2}}}}\) for \(n\) \(=1, 2, 3, ............\) Then,

Which one of the following statements is correct?

Which one of the following statements is correct?

Which one of the following statements is correct?

Which one of the following is a correct statement?

T\(_r\) is always

The sum V\(_1\) + V\(_2\) + ... + V\(_n\) is

Let \(\,{V_r}\) denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r-1). Let $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r =...

Let \(\,{V_r}\) denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r-1). Let $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r =...

Let \({A_1}\), \({G_1}\), \({H_1}\) denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For \(n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}\) have arithmetic, geometric and harminic means ...

Let \({A_1}\), \({G_1}\), \({H_1}\) denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For \(n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}\) have arithmetic, geometric and harminic means ...

Let \({A_1}\), \({G_1}\), \({H_1}\) denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For \(n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}\) have arithmetic, geometric and harminic means ...

Let \(\,{V_r}\) denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r-1). Let $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r =...

In the quadratic equation \(\,\,a{x^2} + bx + c = 0,\) \(\Delta\) \(= {b^2} - 4ac\) and \(\alpha + \beta ,\,{\alpha ^2} + {\beta ^2},\,{\alpha ^3} + {\beta ^3},\) are in G.P. where \(\alpha ,\beta\) are the root of $$\,\,a{x^2} + bx + c...

If total number of runs scored in \(n\) matches is \(\left(\frac{n+1}{4}\right)\left(2^{n+1}-n-2\right)\) where \(n > 1\), and the runs scored in the \(k^{\text {th }}\) match are given by \(k .2^{n+1-k}\), where \(1 \leq k \leq n\). Find, ...

An infinite G.P. has first term '\(x\)' and sum '\(5\)', then \(x\) belongs to

If a, b, c are in A.P., \({a^2}\), \({b^2}\), \({c^2}\) are in H.P., then prove that either a = b = c or a, b, \({ - {c \over 2}}\) form a G.P.

Suppose \(a, b, c\) are in A.P. and \({a^2},{b^2},{c^2}\) are in G.P. If \(a < b < c\) and \(a + b + c = {3 \over 2},\) then the value of \(a\) is

Let a, b be positive real numbers. If a, \({{A_1},{A_2}}\), b are in arithmetic progression, a, \({{G_1},{G_2}}\), b are in geometric progression and a, \({{H_1},{H_2}}\), b are in harmonic progression, show that $$\,{{{G_1},{G_2}} \over {...

Let the positive numbers \(a,b,c,d\) be in A.P. Then \(abc,\) \(abd,\) \(acd,\) \(bcd,\) are

If the sum of the first \(2n\) terms of the A.P.\(2,5,8,......,\) is equal to the sum of the first \(n\) terms of the A.P.\(57,59,61,.....,\) then \(n\) equals

Let \(\alpha\), \(\beta\) be the roots of \({x^2} - x + p = 0\) and \(\gamma ,\delta\) be the roots of \({x^2} - 4x + q = 0.\) If \(\alpha ,\beta ,\gamma ,\delta\) are in G.P., then the integral values of \(p\) and \(q\) respectively, a...

Let \({a_1}\), \({a_2}\),.....,\({a_n}\) be positive real numbers in geometric progression. For each n, let \({A_n}\), \({G_n}\), \({H_n}\) be respectively, the arithmetic mean , geometric mean, and harmonic mean of \({a_1}\),\({a_2}\)........

Consider an infinite geometric series with first term a and common ratio \(r\). If its sum is 4 and the second term is 3/4, then

The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

The harmonic mean of the roots of the equation \(\left( {5 + \sqrt 2 } \right){x^2} - \left( {4 + \sqrt 5 } \right)x + 8 + 2\sqrt 5 = 0\) is

Let \({a_1},{a_2},......{a_{10}}\) be in \(A,\,P,\) and \({h_1},{h_2},......{h_{10}}\) be in H.P. If \({a_1} = {h_1} = 2\) and \({a_{10}} = {h_{10}} = 3,\) then \({a_4}{h_7}\) is

For a positive integer \(n\), let
\(a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}\). Then

Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations
u + 2v + 3w = 6
4u + 5v + 6w = 12
6u + 9v = 4
then show that the roots of the equation \(\left( {{1 \over u} + {1 \over v} + {1 \over w}} \right){x^2}\)...