Sequence of series practice · Mixed difficulty

Let $x_1, x_2, \ldots \in(0, \pi)$ denote the of values of $x$ satisfying the equation $27^{\left(1+|\cos x|+\cos ^2 x+|\cos x|^3+\ldots \text { upto } \infty\right)}=9^3$, find the value of $\frac{1}{\pi}\left(x_1+x_2+\ldots\right)$

If sum of the first 2$n$ terms of an AP series 2, 5, 8,… is equal to the sum of the first $n$ terms of the AP series 57, 59, 61,…, then $n$ equals

If the $p$th term of an AP be $q\;\mathrm{and}\;q$th term be $p$, then its $r$th term will be

If $\tan \left( {\frac{\pi }{9}} \right),\,x,\,\tan \left( {\frac{{7\pi }}{{18}}} \right)$ are in arithmetic progression and $\tan \left( {\frac{\pi }{9}} \right),\,y,\,\tan \left( {\frac{{5\pi }}{{18}}} \right)$ are also in arithmetic progression, then $\left| {x - 2y} \right|$ is equal to

The value of $n$ for which $704+\frac{1}{2}(704)+\frac{1}{4}(704)+\ldots \quad n \quad$ terms $=1984-\frac{1}{2}(1984)+\frac{1}{4}(1984)-\ldots n$ upto terms, is

The sum of the series

upto 15 terms is

A person is to count 4500 currency notes. Let $a_n$ denote the number of notes he counts in the $n$th minute. If $a_1=a_2=\ldots=a_{10}=150$ and $a_{10}, a_{11}, \ldots$ are in AP with common difference - 2, then the time taken by him to count all notes is

If the roots of the equation $x^3-12 x^2+39 x-28=0$ are in AP, then the common difference will be

If $a,b,c$ are in A.P., $b - a,c - b$ and a are in G.P., then $a:b:c$ is

If $9{a^2} + 4{b^2} = 18ab,$ then $\log \left( {3a + 2b} \right) =$

The sum of three numbers in GP is 56 . If we subtract $1,7,21$ from these numbers in that order, we obtain an $A P$. Then, three numbers are

If $x,y,z$ are in H.P., then $\log \left( {x + z} \right) + \log \left( {x - 2y + z} \right)$ is equal to

0.14189189189… can be expressed as a rational number

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

If 7 th and 13 th terms of an AP are 34 and 64 respectively, then its 18 th term is

Let $f\left( x \right)$ be a polynomial function of second degree. If $f\left( 1 \right) = f\left( { - 1} \right)$ and $a,b,c$ are in AP, then $f'\left( a \right),\;f'\left( b \right)$ and $f'\left( c \right)$ are in

If $3^{\text {rd }}, 6^{\text {th }} \&$ last term of HP is $\frac{1}{3}, \frac{1}{5}, \frac{3}{203}$. Then the number of terms is

Let $f:R \to R$ be such that for all $x \in R$ $\left( {{2^{1 + x}} + {2^{1 - x}}} \right),f\left( x \right)$ and $\left( {{3^x} + {3^{ - x}}} \right)$ are in A.P, then the minimum value of $f\left( x \right)$ is

If the sum of an infinite GP and the sum of square of its term is 3, then the common ratio of the first series is

Consider the following statements.

I. A sequence is called an arithmetic progression if the difference of a term and the previous term is always same.

II. Arithmetic Mean (A.M.) $A$ of any two numbers $a$ and $b$ is given by $\frac{1}{2}(a+b)$ such that $a, A, b$ are in A.P. The arithmetic mean for any $n$ positive numbers $a_1, a_2, a_3, \ldots ., a_n$ is given by

$\text {A.M. }=\frac{a_1+a_2+\ldots .+a_n}{n}$

Choose the correct option.

If ${\log _8}x = 2.5$ and ${\log _8}x = 2.5$ then $x =$

The rational number which is equal to the number $2 . \overline{357}$ with recurring decimal is

Let $S = \frac{8}{5} + \frac{{16}}{{15}} + ... + \frac{{128}}{{{2^{18}} + 1}},$, then

The first term of a GP is 7, the last term is 448 and sum of all term is 889, then the common ratio is

The sum of first 9 terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\ldots$ is

If $a,b,c$ be in arithmetic progression, then the value of $\left( {a + 2b - c} \right)\left( {2b + c - a} \right)\left( {a + 2b + c} \right)$ is

Let $a_1, a_2, \ldots, a_9$ be in G.P. with $a_1 < 0$ such that $a_1+a_2=4, a_3+a_4=16$. If $\sum\limits_{i=1}^9 a_i=4 \lambda$, then $\lambda$ is equal to

The sum of infinite terms of the series $\frac{1}{{\left( {1 + a} \right)\left( {2 + a} \right)}} + \frac{1}{{\left( {2 + a} \right)\left( {3 + a} \right)}} + \frac{1}{{\left( {3 + a} \right)\left( {4 + a} \right)}} + ... + \mathrm{to}\;\infty$, where $a$ is a constant, is

If for $x \in \left( {0,\,\frac{\pi }{2}} \right),\,\,{\log _{10}}\sin x + {\log _{10}}\cos x = - 1$ and ${\log _{10}}\left( {\sin x + \cos x} \right) = \frac{1}{2}\left( {{{\log }_{10}}n - 1} \right),\,\,n > 0$, then the value of $n$ is equal to:

Consider an infinite geometric series with first term a and common ratio $r$. If its sum is 4 and the second term is $\frac{3}{4}$, then