Sequence of series practice · Easy

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $\frac{{27}}{{19}}$ Then, the common ration of this series is

The harmonic mean of 4,8 and 16 is

Sum to $n$ terms of the series $1+(1+2)+(1+2+$ 3) $+\ldots$ is

The value of

${\left( {1 + \frac{{{a^2}{x^2}}}{{2\;!}} + \frac{{{a^4}{x^4}}}{{4\;!}} + \ldots } \right)^2} - {\left( {ax + \frac{{{a^3}{x^3}}}{{3\;!}} + \frac{{{a^5}{x^5}}}{{5\;!}} + \ldots } \right)^2},\;\mathrm{is}$

The sum of the first 20 terms common between the series 3 $+7+11+15+\ldots \ldots \ldots$ and $1+6+11+16+\ldots \ldots$, is

If $a,\;b,\;c,\;d$ are any four consecutive coefficients of any expanded binomial, then $\frac{{a + b}}{a},\frac{{b + c}}{b},\frac{{c + d}}{c}$ are in

Sum of $n$ terms to the series $5+7+13+31+85+\ldots \ldots$. , is

$\frac{1}{{2.5}} + \frac{1}{{5.8}} + \frac{1}{{8.11}} + ...\mathrm{upto}\;n$ terms is equal to

If $\log _2\left(5.2^x+1\right), \log _4\left(2^{1-x}+1\right)$ and 1 are in A.P, then $x$ equals

The value of $x=2^{1 / 4} 4^{1 / 8} 8^{1 / 16} 16^{1 / 32} \ldots$ is

The coefficient of $x^{n-2}$ in the polynomial $(x-1)(x-2)(x-3) \ldots(x-n)$, is

If $\frac{{x + y}}{{1 - xy}},y,\frac{{y + z}}{{1 - yz}}$ be in A.P., then $x,\frac{1}{y},z$ will be in

If $1^2+2^2+3^2+\ldots+2009^2= (2009) (4019) (335)$ and

$(1) (2009) + (2) (2008) + (3) (2007) +\ldots+(2009)(1)=(2009)(335)(x)$,

then $x$ equals

Number of value(s) of '$x$' satisfying the equation $\log _2\left(\log _3\left(x^2\right)\right)=1$ is/are:

If $a,b,c$ are in H.P., then the value of $\frac{{b + a}}{{b - a}} + \frac{{b + c}}{{b - c}}$ is

If $\sin \theta, \sqrt{2}(\sin \theta+1), 6 \sin \theta+6$ are in $\mathrm{GP}$, then the fifth term is

The sum of 11 terms of an A.P. whose middle term is 30, is

Let $f(n)=\left[\frac{1}{5}+\frac{3 n}{100}\right] n$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then
$\sum\limits_{{n=1}}^{61} f(n)$

If $1^2+2^2+\ldots+n^2=1015$, then value of $n$ is

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 $\left| a \right| < 1,\left| b \right| < 1,\left| c \right| < 1,\,\,abc \ne 0,$ then

If $\left\{ {{a_n}} \right\}$ is a sequence with ${a_0} = p$ and ${a_n} - {a_{n - 1}} = r{a_{n - 1}}$ for $n \ge 1$, then the terms of the sequence are in

For a sequence $< a_n > , a_1=5$, and $\frac{a_{r+1}}{a_r}=\frac{1}{2} \, \forall \, r$, then $\sum\limits_{{n=1}}^{\infty} a_{2 n-1}$ is

$n$ arithmetic means are inserted between 3 and 17. If the ratio of first to last mean is 1 : 3, then $n$ is
equal to:

150 workers were engaged to finish a piece of work in a certain number of days. 4 workers dropped the second day, 4 more workers dropped the third day and so on. It takes eight more days to finish the work now. The number of days in which the work was completed is

$\sum\limits_{{r=1}}^n r^2-\sum\limits_{{m=1}}^n \sum\limits_{{r=1}}^m r$ is equal to

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

Solution of $0 < |3 x+1| < \frac{1}{3}$ is

$\sum \limits_{k = 1}^\infty \frac{1}{{k!}}\left( {\sum \limits_{n = 1}^\infty {2^{n - 1}}} \right)$ is equal to

If $2\left( {y - a} \right)$ is the H.M. between $y - x$ and $y - z,$ then $x - a,\;y - a,\;z - a$are in

The second term of an AP is $(x-y)$ and the fifth term is $(x+y)$, then its first term is