Sequence of series practice · Mixed difficulty

If $4{a^2} + 9{b^2} + 16{c^2} = 2\left( {3ab + 6bc + 4ca} \right)$, where $a,b,c$ are non-zero numbers, then $a,b,c$ are in

The coefficient of ${x^4}$ in the expansion of $\frac{{1 - 2x - {x^2}}}{{{e^{ - x}}}}$ is

A man saves $₹ 200$ in each of the first three months of his service. In each of the subsequent months his saving increases by $₹ 40$ more than the saving of immediately previous month. His total saving from the start of service will be ₹ 11040 after

If the first, second and the last terms of an A.P. are $a, b$, c respectively, then the sum of the A.P. is

The value of ${\left( {0.16} \right)^{{{\log }_{2.5}}\left( {\frac{1}{3} + \frac{1}{{{3^2}}} + \frac{1}{{{3^3}}} + \ldots \infty } \right)}}$ , is

If the sum of any number of terms in a sequence is always three times the squared number of these terms, then the sequence is

If the $\left( {p + q} \right)$th term of a geometric series is $m$ and the $\left( {p - q} \right)$th term is n, then the $p$th term is

Let $a_1, a_2, \ldots, a_{10}$ be in AP and $h_1, h_2, \ldots, h_{10}$ be in HP. If $a_1=h_1=2$ and $a_{10}=h_{10}=3$, then $a_4 h_7$ is

Let ${a_1},{a_2},{a_3}, \ldots$ be terms of an AP. If $\frac{{{a_1} + {a_2} + \ldots + {a_p}}}{{{a_1} + {a_2} + \ldots + {a_q}}} = \frac{{{p^2}}}{{{q^2}}},\;p \ne q$, then $\frac{{{a_6}}}{{{a_{21}}}}$ equals

The sum to 10 terms of the series $\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\ldots$ is

If $x > 0$, and
$\log _2 x+\log _2(\sqrt{x})+$ $\log _2(\sqrt[4]{x})+\log _2(\sqrt[8]{x})+\log _2(\sqrt[16]{x})+\ldots=4$
then $x$ equals

The first term of a G.P of real numbers given by $b_1, b_2, b_3, \ldots \ldots \ldots$. is Unity. Then the minimum value of $4 b_2+5 b_3$ is $\ldots$.

Let $\frac{1}{x_{1}}, \frac{1}{x_{2}}, \frac{1}{x_{3}}, \ldots .,\left(x_{i} \neq 0\right.$ for $\left.i=1,2, \ldots, n\right)$ be in A.P. such that $x_{1}=4$ and $x_{21}=20$. If $n$ is the least positive integer for which $x_{n}>50$, then $\sum_{i=1}^{n}\left(\frac{1}{x_{i}}\right)$ is equal to.

If $a,b,c$ are in G.P. and ${\log _c}a,{\log _b}c,{\log _a}b$ are in A.P., then the common difference of the A.P. is

If $a, b, c$ are in AP and $(a+2 b-c)(2 b+c-a)(c+a-b)$ $=\lambda a b c$, then $\lambda$ is

The series expansion of $\log \left\{ {{{\left( {1 + x} \right)}^{1 + x}}{{\left( {1 - x} \right)}^{1 - x}}} \right\},$ is

If ${\log _7}\left\{ {{{\log }_5}\left( {\sqrt {x + 5} + \sqrt x } \right)} \right\} = 0$ then $x =$

If $2, a, 2^3, b$ and $2^5$ are in $G P$, then $a b$ is

Fifth term of a G.P. is 2, then the product of its 9 terms is

If $a, b, c$, are in A.P. and $p, p^{\prime}$ are respectively A.M. and GM. between $a$ and $b$ while $q, q^{\prime}$ are respectively AM and $\mathrm{GM}$, between $\mathrm{b}$ and $\mathrm{c}$, then

In an AP the sum of any two terms, such that the distance of one of them from the beginning is same as that of the other from the end, is

The sum of $n$ arithmetic means between $a$ and $b$ is

Let $a_1, a_2, \ldots, a_{10}$ be in AP and $h_1, h_2, \ldots, h_{10}$ be in HP. If $a_1=h_1=2$ and $a_{10}=h_{10}=3$, then $a_4 h_7$ is

If $a_1, a_2, a_3, \ldots$ are in GP with first term $a$ and common ratio $r$, then $\frac{a_1 a_2}{a_1^2-a_2^2}+\frac{a_2 a_3}{a_2^2-a_3^2}+\frac{a_3 a_4}{a_3^2-a_4^2}+\ldots+\frac{a_{n-1} a_n}{a_{n-1}^2-a_n^2}$ is equal to

The first 3 terms of the sequence which is defined by $a_n=\frac{n}{n^2+1}$, are

The value of $\sum_{\mathrm{r}=16}^{30}(\mathrm{r}+2)(\mathrm{r}-3)$ is equal to :

If $a$ and $b$ are two different positive real numbers, then which of the following statements is true?

The sum of the series ${1.3^2} + {2.5^2} + {3.7^2} + \;...$ upto 20 terms is

The first and second terms of a G.P. are $x^{-4}$ and $x^n$ respectively. If $x^{52}$ is the eight th term of the same progression, then $n$ is equal to

The value of $\sqrt {\log _{0.5}^24} ,$ is