Sequence of series practice · Mixed difficulty

Let $S \subset(0, \pi)$ denote the set of values of $x$ satisfying the equation
${8^{1 + |\cos x| + {{\cos }^2}x + \left| {{{\cos }^3}x} \right| + \ldots \mathrm{ to }\infty }} = {4^3}$
Then $S=$

If the sum of first 11 terms of an A.P. $a_1, a_2, a_3, \ldots$ is 0 , and the sum of the A.P. $a_1, a_3, a_5, \ldots, a_{23}$ is $k a_1, a_1 \neq 0$. Then $k$ is equal to

Let ${T_r}$, be $r$th term of an AP whose first term is $a$ and common difference is $d$. If for some positive integers $m,n,m \ne n,\;{T_m} = \frac{1}{n}$ and ${T_n} = \frac{1}{m}$, then $a - d$ equals

If $a,\;b,\;c$ are digits, then the rational number represented by 0. $cababab \ldots$ is

If sum of $n$ terms of an AP is $2n + 3{n^2}$ then $r$th term is

If $\left| x \right| < 1$ and $y = x - \frac{{{x^2}}}{2} + \frac{{{x^3}}}{3} - \frac{{{x^4}}}{4} + ...$, then $x$ is equal to

If sum of first 20 terms of an A.P. is equal
to sum of first 30 terms of the A.P. then sum of the first 50
terms of the A.P. is

The sum to $n$ terms of the series $\frac{1}{1^3}+\frac{1+2}{1^3+2^3}+\frac{1+2+3}{1^3+2^3+3^3}+\ldots$ is

$\frac{{{{\log }_2}a}}{3} = \frac{{{{\log }_2}b}}{4} = \frac{{{{\log }_2}c}}{{5\mathrm{\lambda }}}$ $and$ ${a^{ - 3}}{b^{ - 4}}c = 1$ then $\mathrm{\lambda } =$

The sum of the integers from 1 to 100 which are not divisible by 3 or 5 is

If the AM and GM of roots of a quadratic equations are 8 and 5 respectively, then the quadratic equation will be

If $\frac{{{e^{5x}} + {e^x}}}{{{e^{3x}}}}$ is expand in a series of ascending powers of $x$ and $n$ is an odd natural number, then the coefficient of ${x^n},$ is

The value of ${3^{\frac{4}{{{{\log }_4}9}}}} + {27^{\frac{1}{{{{\log }_{36}}9}}}} + {81^{\frac{1}{{{{\log }_5}3}}}},\;\mathrm{is}$

Sum of the series $\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+\ldots \ldots$ upto $n$ terms is equal to

If $x_1, x_2, x_3$ as well as $y_1, y_2, y_3$ are in GP with same common ratio, then the points $\left(x_1, y_1\right),\left(x_2, y_2\right)$ and $\left(x_3, y_3\right)$

If $2^{1-x}+2^{1+x}, f(x), 3^x+3^{-x}$ are in A.P., then the minimum value of $f(x)$ is

If the lengths of sides of a right triangle are in AP, then the sines of the acute angle are

Let $s_{1}, s_{2}, s_{3} \ldots, s_{10}$ respectively be the sum of 12 terms of 10 A.Ps whose first terms are $1,2,3, \ldots ., 10$ and the common differences are $1,3,5, \ldots, 19$ respectively. Then $\sum\limits_{{i=1}}^{10} s_{i}$ is equal to

If a geometric progression consisting of positive terms, each term equals the sum of the next two terms, then the common ratio of this progression equals

The length of a side of a square is $a$ metre. A second square is formed by joining the mid-points of these square. Then, a third square is formed by joining the mid-points of the second square and so on. Then, the sum of the area of the squares which carried upto infinity, is

If $S_r$ denotes the sum of the first $r$ terms of an $A P$, then $\frac{S_{3 r}-S_{r-1}}{S_{2 r}-S_{2 r-1}}$ is equal to

The sum of all natural numbers $'n'$ such that $100\, < \,n < 200$ and HCF $(91,\,n)\, > 1$ is

If $3^{2 \sin 2 \theta-1}, 14,3^{4-2 \sin 2 \theta}$ are first three terms of an A.P., then its fifth term is given by

The sum of the series $1 + \frac{{{1^2} + {2^2}}}{{2\;!}} + \frac{{{1^2} + {2^2} + {3^2}}}{{3\;!}} + \frac{{{1^2} + {2^2} + {3^2} + {4^2}}}{{4\;!}} + \ldots ,$ is

If $(1.05)^{50}=11.658$, then $\sum_{n=1}^{49}(1.05)^n$ equals

If the product of $n$ positive numbers is unity, then their sum is

Let $a$ and $b$ be two given numbers. If $A$ denotes the single AM and $S$ denotes the sum of $n$ AM's between $a$ and $b$, then $S / A$ depends on

If ${a_1},{a_2},{a_3},{a_4},{a_5}$ and ${a_6}$ are six arithmetic means between 3 and 31, then ${a_6} - {a_5}$ and ${a_1} + {a_6}$ are respectively equals to

$\sum\limits_{{j=1}}^{n} \sum\limits_{{i=1}}^{n} i$ is equal to

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 alternativety positive and negative, then the first term is