Sequence of series practice · Medium

Ankur appeared in an examination which has 5 subjects, out of five, in four subjects he got $90,70,75,65$ marks respectively. The minimum & maximum marks he should score in fifth subject so that the average mark is greater than or equal to 70 and less than or equal to 75 is

The sum of the first 20 terms of the series $1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\ldots$ is?

If $(10)^9+2(11)^1(10)^8+3(11)^2(10)^7+\ldots+(10)(11)^9$ $=k(10)^9$, then $k$ is equal to

Let $a, b, c \in R$. If $f(x)=x^{2}+b x+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+x y, \forall x, y \in R$, then $\sum_{n=1}^{10} f(n)$ is equal to:

Let $a,b,c$ are in GP and $4a,5b,4c$ are in AP such that $a + b + c = 70$, then value of $b$ is

If $\frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ is the harmonic mean between $a$ and $b$, then the value of $n$ is

The AM, GM and HM of two numbers are $x, y$ and $z$, respectively. Then, which of the following is true?

If $a, b, c$ are three distinct real numbers are in G.P and $a+b+c=x b$ then Which of the following statements are true

Statement-I : $x$ can not take the values between $-1$ and 2

Statement-II : The least positive integral value of $x$ is 3

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

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

The sum of $n$ terms of the following series $1.2+2.3+3.4+4.5+\ldots$ shall be

Let $S_{n}=\frac{1}{1^{3}}+\frac{1+2}{1^{3}+2^{3}}+\frac{1+2+3}{1^{3}+2^{3}+3^{3}}+\ldots$. $+\frac{1+2+\ldots \ldots+n}{1^{3}+2^{3}+\ldots \ldots+n^{3}}$, If $100 \mathrm{S}_{n}=n$, then $\mathrm{n}$ is equal to:

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

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

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

If $x$ is added to each of numbers $3,9,21$ so that the resulting numbers may be in GP, then the value of $x$ will be

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

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

If $a_1, a_2 \ldots, a_n$ are positive real numbers whose product is a fixed number $\mathrm{c}$, then the minimum value of $a_1+a_2+\ldots \ldots+a_{n-1}+2 a_n$ is

If $\log (x+z)+\log (x+z-2 y)=2 \log (x-z)$, then $x, y$ and $z$ are in

If $x=\sqrt[3]{(\sqrt{2}+1)}-\sqrt[3]{(\sqrt{2}-1)}$; then $x^{3}+3 x=$

If $\cos (x-y), \cos x$ and $\cos (x+y)$ are in HP, then $\cos x \sec \left(\frac{y}{2}\right)$ is equal to

$\sin ^{-1}\left(a-\frac{a^2}{3}+\frac{a^3}{9}+\ldots.\right)+\cos ^{-1}\left(1+b+b^2+\ldots\right)=\frac{\pi}{2}$ when

If ${3^{2x + 1}}{.4^{x - 1}} = 36,$ then $x =$

The sum of the series $1 + 3x + 6{x^2} + 10{x^3} + ...\infty$ will be

If $\log _{10}\left(\frac{1}{2^x+x-1}\right)=x\left(\log _{10} 5-1\right)$ then $x$ is equal to