Sequence and Series

If $t_n$ denotes the $n^{\text {th }}$ term of an A.P. and $t_p=\frac{1}{q}, t_q=\frac{1}{p}$, then which one of the following options is a root of the equation $(p+2 q-3 r) x^2+(q+2 x-3 p) x+(r+2 p-3 q)=0 ?$

Let $a_1, a_2, a_3 \ldots$ are in G.P. such that $n>m, a_n>a_m$ and $a_1+a_n=66, a_2 \cdot a_{n-1}=128$. If $\sum_{r=1}^n a_r=126$, then $n$ is

If the sum of ' $n$ ' terms of an A.P. is $3 n^2+5 n$ and its $m$ th term is 164 , then the value of $m$ is

If $a, b, c$ are in A.P. and if the equations $(b-c) x^2+(c-a) x+(a-b)=0$ and $2(c+a) x^2+(b+c) x=0$ have a common root, then

The sum of the first four terms of an arithmetic progression is 56 . The sum of the last four terms is 112. If its first term is 11, then the number of terms is

If \(\alpha_1, \alpha_2, \ldots, \alpha_n\) are in A.P. with common difference \(\theta\), then the sum of the series
$$
\sec \alpha_1 \sec \alpha_2+\sec \alpha_2 \sec \alpha_3+\ldots .+\sec \alpha_{n-1} \sec \alpha_n=k\left(\tan \alpha_n-\...

If for the series \(a_1, a_2, a_3\), ...... etc, \(\mathrm{a}_{\mathrm{r}}-\mathrm{a}_{\mathrm{r}+\mathrm{i}}\) bears a constant ratio with \(\mathrm{a}_{\mathrm{r}} \cdot \mathrm{a}_{\mathrm{r}+1}\); then $$\mathrm{a}_1, \mathrm{a}_2, \mat...

Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If \(a_n\) and \(b_n\) be the \(n^{\text {th }}\) term of A.P. and G.P. respectively then

Let \({a_1},{a_2},{a_3},\,...,\,{a_n}\) be positive real numbers. Then the minimum value of \({{{a_1}} \over {{a_2}}} + {{{a_2}} \over {{a_3}}}\, + \,...\, + \,{{{a_n}} \over {{a_1}}}\) is

Consider a quadratic equation \(a{x^2} + 2bx + c = 0\) where a, b, c are positive real numbers. If the equation has no real root, then which of the following is true?

If \(1,{\log _9}({3^{1 - x}} + 2),{\log _3}({4.3^x} - 1)\) are in A.P., then x equals

If the n terms \({a_1},{a_2},\,......,\,{a_n}\) are in A.P. with increment r, then the difference between the mean of their squares & the square of their mean is

Let \({a_n} = {({1^2} + {2^2} + .....\,{n^2})^n}\) and \({b_n} = {n^n}(n!)\). Then

If a, b, c are in G.P. and log a \(-\) log 2b, log 2b \(-\) log 3c, log 3c \(-\) log a are in A.P., then a, b, c are the lengths of the sides of a triangle which is

Three unequal positive numbers a, b, c are such that a, b, c are in G.P. while \(\log \left( {{{5c} \over {2a}}} \right),\log \left( {{{7b} \over {5c}}} \right),\log \left( {{{2a} \over {7b}}} \right)\) are in A.P. Then a, b, c are the leng...

The digit in the unit's place of the number 1! + 2! + 3! + .... + 99! is

Consider the real valued function h : {0, 1, 2, ...... 100} \(\to\) R such that h(0) = 5, h(100) = 20 and satisfying h(p) = \({1 \over 2}\) {h(p + 1) + h(p \(-\) 1)} for every p = 1, 2 ..... 99. Then the value of h(1) is

Let a, b, c be real numbers, each greater than 1, such that \({2 \over 3}{\log _b}a + {3 \over 5}{\log _c}b + {5 \over 2}{\log _a}c = 3\). If the value of b is 9, then the value of 'a' must be

Let I(n) = nn, J(n) = 13.5 ......... (2n \(-\) 1) for all (n > 1), n \(\in\) N, then

If a and b are arbitrary positive real numbers, then the least possible value of \({{6a} \over {5b}} + {{10b} \over {3a}}\) is

In a certain test, there are n questions. In this test 2n-i students gave wrong answers to at least i questions, where i = 1, 2, ..., n. If the total number of wrong answers given is 2047, then n is equal to

Let x1, x2 be the roots of \({x^2} - 3x + a = 0\) and x3, x4 be the roots of \({x^2} - 12x + b = 0\). If \({x_1} < {x_2} < {x_3} < {x_4}\) and \({x_1},{x_2},{x_3},{x_4}\) are in GP, then ab equals

Given that n numbers of arithmetic means are inserted between two sets of numbers a, 2b and 2a, b where a, b \(\in\) R. Suppose further that the mth means between these sets of numbers are same, then the ratio a : b equals

In a GP series consisting of positive terms, each term is equal to the sum of next two terms. Then, the common ratio of this GP series is

The sum of n terms of the following series \({1^3} + {3^3} + {5^3} + {7^3} + ...\) is

If a, x are real numbers and | a | < 1, | x | < 1, then 1 + (1 + a) x + (1 + a + a2) x2 + ..... \(\infty\) is equal to

The value of \(\sum\limits_{r = 2}^\infty {{{1 + 2 + .... + (r - 1)} \over {r!}}}\)

The harmonic mean of two numbers is 4. Their arithmetic mean A and the geometric mean G satisfy the relation 2A + G2 = 27. Find the numbers.

The value of \({2 \over {3!}} + {4 \over {5!}} + {6 \over {7!}} +\) ............ is

If angles A, B and C are in A.P., then \({{a + c} \over b}\) is equal to

The value of n for which \({{{x^{n + 1}} + {y^{n + 1}}} \over {{x^n} + {y^n}}}\) is the geometric mean of x and y is

G.M. and H.M. of two numbers are 10 and 8 respectively. The numbers are

Sum of n terms of the following series 13 + 33 + 53 + 73 ......... is

If sum of an infinite geometric series is \({4 \over {3}}\) and its 1st term is \({3 \over {4}}\), then its common ratio is

Find the sum of the first n terms of the series 0.2 + 0.22 + 0.222 + ......

The sum of the infinite series \(1 + {1 \over {2!}} + {{1\,.\,3} \over {4!}} + {{1\,.\,3\,.\,5} \over {6!}} + \,....\) is

The coefficient of xn, where n is any positive integer, in the expansion of (1 + 2x + 3x2 + ....... \(\infty\))1/2 is

For what value of m, \({{{a^{m + 1}} + {b^{m + 1}}} \over {{a^m} + {b^m}}}\) is the arithmetic mean of a and b?

If three positive real numbers a, b, c are in A.P. and abc = 4 then the minimum possible value of b is

If a, b, c are G.P. (a > 1, b > 1, c > 1), then for any real number x (with x > 0, x \(\ne\) 1) logax, logbx, logcx are in

The sum of the infinite series \({\left( {{1 \over 3}} \right)^2} + {1 \over 3}{\left( {{1 \over 3}} \right)^4} + {1 \over 5}{\left( {{1 \over 3}} \right)^6} + ...\) is

If three real numbers a, b, c are in Harmonic Progression, then which of the following is true?

If a, b, c be in Arithmetic progression, then the value of \((a + 2b - c)(2b + c - a)(a + 2b + c)\) is

The 5th term of the series \({{10} \over 9},{1 \over 3}\sqrt {{{20} \over 3}} ,{2 \over 3}\) is

The sum of the series \((1 + 2) + (1 + 2 + {2^2}) + (1 + 2 + {2^2} + {2^3}) + ....\) upto n terms is

If the mth term and the nth term of an A.P. are respectively \({1 \over n}\) and \({1 \over m}\), then the (mn)th term of the A.P. is