Sequence of series practice · Easy

If ${a^x} = {b^y} = {c^z} = {d^w},$ then ${\log _a}\left( {bcd} \right)$ equals to

An A.P., a G.P. and a H.P. have the same first and last terms and the same odd number of terms. The middle terms of the three series are in

If $< {a_n} >$ is an arithmetic sequence, then $\Delta \; = \left| {\begin{array}{cccccccccccccccccccc} {{a_m}}&{{a_n}}&{{a_p}}\\ m&n&p\\ 1&1&1 \end{array}} \right|$ equals

If $x,\;1,\;z$ are in AP and $x,\;2,\;z$ are in GP, then $x,\;4,\;z$ will be in

Let $f(x)$ be a polynomial of second degree. If $f(1)=f(-1)$ and $a, b$, $c$ are in A.P., then $f^{\prime}(a), f^{\prime}(b)$ and $f^{\prime}(c)$ are in

The least value of ' $a$ ' for which $5^{1+x}+5^{1-x}, a / 2,25^x+25^{-x}$ are three consecutive terms of an A.P., is

The value of the sum $\sum \limits_{r = 1}^n \sum \limits_{s = 1}^n {S_{rs}}\;{2^r}{3^s},\mathrm{where}{S_{rs}} = 0,\;\mathrm{if}\;r \ne s\;\mathrm{and}{S_{rs}} = 1,\;\;\mathrm{if}\;r = s,$ is

If $S=1+a+a^2+\ldots \ldots$ to $\infty(a<1)$, then the value of $a$, is

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

If $\left| x \right| < 1$ then the sum of the series $1 + 2x + 3{x^2} + 4{x^3} + ...\infty$ will be

Consider the following statements

I. If ${a}_1, {a}_2, \ldots, {a}_{{n}} \ldots$ is a sequence, then the expression $a_1+a_2+\ldots+a_n+\ldots$ is called a series.

II. Those sequences whose terms follow certain patterns are called progressions.

Choose the correct option.

Suppose all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is $6: 11$ and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

If $3 + {\log _5}x = 2{\log _{25}}y,$then $x$ equals to

If $A_1, A_2$ be two A.M.'s and $G_1, G_2$ be two G.M.'s between $a$ and $b$, then $\frac{A_1+A_2}{G_1 G_2}$ is equal to

Let $a$ be a positive number such that the arithmetic mean of $a$ and 2 exceeds their geometric mean by 1. Then, the value of $a$ is

Let $b_i>1$ for $i=1,2, \ldots, 101$. Suppose $\log _e b_1$, $\log _e b_2, \ldots, \log _e b_{101}$ are in Arithmetic Progression (A.P) with the common difference${\log _e}2$ . Suppose $a_1, a_2, \ldots, a_{101}$ are in A.P. such that $a_1=b_1$ and $a_{51}=b_{51}$. If $t=b_1+b_2+\ldots+b_{51}$ and $s=a_1+a_2+\ldots+a_{51}$, then

If $1 + \cos \alpha + {\cos ^2}\alpha + ...\infty = 2 - \sqrt 2 ,$ then $\alpha ,\;(0 < \alpha < \pi )$ is

If $d,\;e,\;f$ are in G.P. and the two quadratic equations $a{x^2} + 2\;bx + c = 0$ and $d{x^2} + 2\;ex + f = 0$ have a common root, then

If $p^{t h}, q^{\text {th }}$ and $r^{\text {th }}$ terms of A.P. $a_1, a_2, a_3$, are in G.P., then $a_p: a_q=$

The value of $\sum\limits_{r = 1}^\infty {\frac{{(4r + 5){5^{ - r}}}}{{r(5r + 5)}}}$ is

Let ${a_1},{a_2},{a_3},...$ be an AP with ${a_6} = 2$. Then, the common difference of this AP, which maximizes the product ${a_1},{a_4},{a_5},$ is

The third term of a geometric progression is 4 . The product of the first five terms is

The arithmetic mean between two numbers is $A$, and $S$ is the sum of $n$ arithmetic means between these mumbers, then

The sum to infinity of the series, $1+2\left(1-\frac{1}{n}\right)+3\left(1-\frac{1}{n}\right)^2+\ldots$ is

Let $T_r$ be the $r^{\text {th }}$ term of an A.P. whose first term is a and common difference is $d$. If for some positive integers $m, n$, $m \neq n, T_m=\frac{1}{n}$ and $T_n=\frac{1}{m}$, then $a-d$ equals

If $x = {\log _a}\left( {bc} \right),y = {\log _b}\left( {ca} \right)$ and $z = {\log _c}\left( {ab} \right),$ then which of the following is correct?

Let $S = 2 + \frac{6}{7} + \frac{{12}}{{{7^2}}} + \frac{{20}}{{{7^3}}} + \frac{{30}}{{{7^4}}} + \ldots$ Then $4 S$ is equal to

If $a, b, c$ are in H.P. then $a-\frac{b}{2}, \frac{b}{2}, c-\frac{b}{2}$ are in

If a is positive and if $A$ and $G$ are the arithmetic mean and the geometric mean of the roots of ${x^2} - 2ax + {a^2} = 0$ respectively, then

Three numbers are in AP such that their sum is 18 and sum of their squares is 158. The greatest number among them is