Sequence of series practice · Mixed difficulty

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 $n$th term of a GP is 128 and the sum to its $n$ terms is 255 . If its common ratio is 2 , then its first term is

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

If $S = \sum \limits_{n = 2}^\infty {\;^n}{C_2}\frac{{{3^{n - 2}}}}{{n\;!}},$ then $S$ equals

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 ${S_n} = \frac{1}{{6.11}} + \frac{1}{{11.16}} + \frac{1}{{16.21}} + ...\mathrm{to}\;n$ terms, then $6{S_n}$ equals

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

If $a, b, c$ and $d$ are in HP, then $a b+b c+c d$ is equal to

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

If it is given that $\frac{1}{{{1^4}}} + \frac{1}{{{2^4}}} + \frac{1}{{{3^4}}} + \ldots \mathrm{to }\infty = \frac{{{\mathrm{\pi }^4}}}{{90}},$, then the value of $\frac{1}{{{1^4}}} + \frac{1}{{{3^4}}} + \frac{1}{{{5^4}}} + \ldots \mathrm{to }\infty$ is equal to

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

If $a, b$ and $c$ are in $A P$, then $a+\frac{1}{b c}, b+\frac{1}{c a}$ and $c+\frac{1}{a b}$ are in

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

If ${S_1},{S_2},{S_3}$ be the sum of $n,2n,3n$ $n$ terms respectively of an A.P., then

The value of ${\log _{\sqrt 2 }}\sqrt {2\sqrt {2\sqrt {2\sqrt 2 } } } \;,\;\mathrm{is}$

The sum upto $\left( {2n + 1} \right)$ terms of the series ${a^2} - {\left( {a + d} \right)^2} + {\left( {a + 2d} \right)^2} - {\left( {a + 3d} \right)^2} + \ldots ,$ is

If $a,b,c$ are in A.P., then $a + \frac{1}{{bc}},b + \frac{1}{{ca}},c + \frac{1}{{ab}}$ are in

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

If $\frac{1}{a}, \frac{1}{b}$ and $\frac{1}{c}$ are in AP, then $\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)$ $\left(\frac{1}{b}+\frac{1}{c}-\frac{1}{a}\right)$ is equal to

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

If $S_n$ denotes the sum of first $n$ term of an A.P., then $\frac{S_{3 n}-S_{n-1}}{S_{2 n}-S_{2 n-1}}$ is equal to

In an A.P., the first term is 1 and sum of the first $p$ terms is 0, then sum of the first $(p+q)$ terms is

Suppose $a_1, a_2, \ldots a_n, \ldots$ be an arithmetic progression of natural numbers. If the ration of the sum of first five terms to the sum of first nine terms of the progression is $5: 17$ and $110\, < \,{a_{15}}\, < \,120$, then the sum of the first ten terms of the progression is equal to