Sequence of series practice · Mixed difficulty

If $\log _{12} 27=a$, then $\log _6 16=$

Let the sequence, ${a_1},{a_2},{a_3}, \ldots ,\;{a_{2n}},$ form an AP, then $a_1^2 - a_2^2 + a_3^2 - ... + a_{2n - 1}^2 - a_{2n}^2$ is equal to

Let $a$, $b$ and $c$ be in GP with common ratio $r$, where $a \ne 0$ and $0 < r \le \frac{1}{2}$. If 3a, 7b and 15c are the first three terms of an AP, then the 4th term of this AP is

If one geometric mean $G$ and two arithmetic means $A_1$ and $A_2$ are inserted between two given quantities, then $\left(2 A_1-A_2\right)\left(2 A_2-A_1\right)=$

The number of numbers lying between 100 and 500 that are divisible by 7 but not by 21 , is

If $x_n>x_{n-1}>\ldots>x_2>x_1>1,$ the value of ${\log _{{x_1}}}\,{\log _{{x_2}}}\,{\log _{{x_3}}}\, \cdot \cdot \cdot {\log _{{x_n}}}\,{x_n}^{{x_{n - 1}}^{{.^{{.^{{.^{{.^{.{x_1}}}}}}}}}}}$ is

If $1^2+2^2+3^2+\ldots+2003^2=(2003)(4007)(334)$ and $(1)(2003)+(2)(2002)+(3)(2001)$
$+\ldots+(2003)(1)=(2003)(334)(x)$
then $x$ equals

The solution of the equation $\log _7 \log _5(\sqrt{x+5}+\sqrt{x})=0$ is

If $a_1, a_2, a_3\left(a_1>0\right)$ are three successive terms of a G.P. with common ratio $r$, the value of for which $a_3>4 a_2-3 a_1$ holds is given by

If $x_{1}, x_{2}, \ldots ., x_{n}$ and $\frac{1}{h_{1}}, \frac{1}{h^{2}}, \ldots . \frac{1}{h_{n}}$ are two A.P's such that $x_{3}=h_{2}=8$ and $x_{8}=h_{7}=20$, then $x_{5} . h_{10}$ equals.

A polygon has 25 sides, the lengths of which starting from the smallest side are in AP. If the perimeter of the polygon is $2100 \mathrm{~cm}$ and the length of the largest side 20 times that of the smallest, then the length of the smallest side and the common difference of an AP are

The value of $49^A+5^B$, where $A=1-\log _7 2$ and $B=-\log _5 4$ is

After inserting $n$ A.Ms between 2 and 38, the sum of the resulting progression is 200. The value of $n$ is

${e^{x - 1 - \frac{1}{2}{{\left( {x - 1} \right)}^2} + \frac{1}{3}{{\left( {x - 1} \right)}^3} - \frac{1}{4}{{\left( {x - 1} \right)}^4} + \ldots }}$ is equal to

Let the sum of the first $\mathrm{n}$ terms of a non-constant A.P., $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3, \ldots$. be $50 \mathrm{n}+\frac{\mathrm{n}(\mathrm{n}-7)}{2} \mathrm{~A}$, where $\mathrm{A}$ is a constant. If $\mathrm{d}$ is the common difference of this A.P., then the ordered pair $\left(\mathrm{d}, \mathrm{a}_{50}\right)$ is equal to

The value of $\sum\limits_{{r=1}}^n \frac{1}{\sqrt{a+r x}+\sqrt{a+(r-1) x}} \text { is }$

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 in the immediately previous month. His total savings from the start of service will be $₹ 11040$ after

If the sum of $n$ terms of an $A P$ is $3 n^2+5 n$, then the common difference is

If ${a_n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \ldots + \frac{1}{{{2^n} - 1}},$ then

If $a,b,c$ are in G.P. and $a - b,c - a$ and $b - c$ are in H.P., then $a + 4b + c$ is equal to

Suppose $a, b, c$ are distinct positive real numbers such that $a, 2 b, 3 c$ are in A.P. and $a, b, c$ are in G.P. The common ratio of G.P. is

If one geometric mean $G$ and two arithmetic means $p$ and $q$ are inserted between two numbers, then $G^2$ is equal to

The sum of 7 terms of a GP $3,6,12, \ldots$ is

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12 , then its $4^{\text {th }}$ term is:

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

The four numbers are in AP, such that their sum is 50 and greatest of them is 4 times the least, are

If $1 + \sin x + {\sin ^2}x + ...\mathrm{upto}\;\infty = 4 + 2\sqrt 3 ,0 < x < \pi \;\mathrm{and}\;x \ne \frac{\pi }{2}$, then $x$ is equal to

Let $a$ and $b$ be roots of ${x^2} - 3\;x + p = 0$ and let $c$ and $d$ be the roots of ${x^2} - 12\;x + q = 0,$ where $a,b,c,d$ form an increasing GP Then the ratio of $q + p:q - p$ is equal to

Statement I: If '$a$' is the first term and '$d$' is the common difference of an A.P., then its $n^{\text {th }}$ term is given by $a_n=a-(n-1) d$

Statement II: The sum $S_n$ of $n$ terms of an A.P. with first term '$a$' and common difference '${d}$' is given by ${S}_{{n}}=\frac{{n}}{2}\{2 {a}+({n}-1) {d}\}.$

Choose the correct option.

If $a,\;b,\;c$ are in AP, then ${3^a},\;{3^b},\;{3^c}$ shall be in