Sequence of series practice · Easy

If $a, b, c$ be respectively the $p^{t h}$, $q^{\text {th }}$ and $r^{th}$ terms of a G.P., then

$\Delta=\left|\begin{array}{ccc} \log a & \log b & \log c \\ p & q & r \\ 1 & 1 & 1 \end{array}\right| \text { equals }$

The first term of a GP $a_1, a_2, a_3, \ldots$ is unity. The value of $4 a_2+5 a_3$ is minimum when the common ratio is

If AM of two numbers is twice of their GM, then the ratio of greatest number to smallest number is

$11^2+12^2+13^2+\ldots \ldots20^2=$

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

In an increasing arithmetic progression, the sum of the first three terms is 27 and the sum of their squares is 275 . The common difference of the AP is

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 $\left(2 A_1-A_2\right)\left(2 A_2-A_1\right)=$

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

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.

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

If $b+c, c+a, a+b$ are in H.P., then $\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{a+b}$ are in

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

If $\mathrm{A}$ is the arithmetic mean of $a$ and $b$ and $p$ and $q$ are two geometric means between $a$ and $b$, then $2 \mathrm{A}=$

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

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

Let $S = \sum\limits_{r = 1}^{117} {\frac{1}{{2[\sqrt r ] + 1}}} ,$ when $[\cdot]$ denotes the greatest integer function and if $S=\frac{p}{q}$, when $p$ and $q$ are co-primes, the value of $p+q$ is

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

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

The value of $4^{1 / 3} \cdot 4^{1 / 9} \cdot 4^{1 / 27} \ldots \infty$ is

If ${\log _2}\sin x - {\log _2}\cos x - {\log _2}\left( {1 - {{\tan }^2}x} \right) = - 1,$ then $x =$

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

Let $a_1, a_2, \ldots a_n, \ldots$ be a G.P. of positive terms such that $\sum\limits_{n=1}^{100} a_{2 n+1}=200, \sum\limits_{n=1}^{100} a_{2 n}=100$. Then, $\sum\limits_{n=1}^{200} a_n$ is equal to

If $a, b, c$, d are in H.P., then