Sequence of series practice · Medium

If $p, q, r$ are in $A P$ and $x, y, z$ are in GP, then $x^{q-r} \cdot y^{r-p} \cdot z^{p-q}$ is equal to

For a given arithmetic series the sum of first 50 terms is 200 and the sum of the next 50 terms is 2700, the first term of the series is

Let $a_1, a_2, a_3 \ldots a_n$ be in A.P. and $S_n$ denotes the sum of first $n$ terms of this A.P. If $S_{10}=390, \frac{a_{10}}{a_{50}}=\frac{15}{7}$, then $S_{15} - S_5$ is equal to

In a GP, if $(m+n)$ th term is $p$ and $(m-n)$ th term is $q$, then its $m$ th term is

$\frac{1}{{1 \cdot 2 \cdot 3 \cdot 4}} + \frac{4}{{3 \cdot 4 \cdot 5 \cdot 6}} + \frac{9}{{5 \cdot 6 \cdot 7 \cdot 8}} + \frac{{16}}{{7 \cdot 8 \cdot 9 \cdot 10}} + \ldots \mathrm{to\;}\infty$ is equal to

Let $3, a, b, c$ be in A.P. and $3, a-1, b+1, c+9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is:

The sum of the series $\left( {1 + 2} \right) + \left( {1 + 2 + {2^2}} \right) + \left( {1 + 2 + {2^2} + {2^3}} \right) + ...\mathrm{upto}\;n$ terms is

If $p\mathrm{th},q\mathrm{th},r\mathrm{th}$ and sth terms of an A.P. are in G.P., then $p - q,q - r,r - s$ are in

The coefficients of ${x^n}$ in the expansion of ${\log _a}\left( {1 + x} \right)$ is

If $x,y,z$ are positive integers, then $\left( {x + y} \right)\left( {y + z} \right)\left( {z + x} \right)$ is

The sixth term of an A.P. is equal to 2 the value of the common difference of the A.P. which makes the product ${a_1}{a_4}{a_5}$ least is given by

The harmonic mean of two numbers is 4 and the arithmetic and geometric means satisfy the relation $2\;A + {G^2} = 27,$ the numbers are

If $p, q, r$ are in one geometric progression and $a, b, c$ in another geometric progression, then $c p, b q, ar$ are in

The sum of the series : $^{2}+2(d)^{2}+3(6)^{2}+\ldots$ upto 10 terms is :

Consider the sequence $8 A+2 B, 6 A+B, 4 A, 2 A-B$, which term of this sequence will have a coefficient of $A$ which is twice the coefficient of $B$ ?

The value of $S=\sum\limits_{{k=0}}^n \frac{(-1)^k}{k^3+9 k^2+26 k+24} \cdot{ }^n C_k$ is

If $a, b$ and $c$ are in HP, then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$ always passes through a fixed point and that point is

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x-1= 0$ and $S_n=2024 \alpha^n+2024 \beta^n$ then $S_3$ is equal to

The consecutive numbers of a three digit number form a GP. If we subtract 792 from this number, then we get a number consisting of the same digits written in the reverse order and if we increase the second digit of the required number by 2 , then resulting number forms an AP. The number is

Let ${S}_1=\sum\limits_{{j}=1}^{10} {j}({j}-1)^{10} \mathrm{C}_{{j}}, {S}_2=\sum\limits_{{j}=1}^{10} {j}^{10} \mathrm{C}_{{j}}$ and ${S}_3=\sum\limits_{{j}=1}^{10} {j}^{2} \, {}^{10}\mathrm{C}_{{j}}$. Consider

Statement -1: $S_3=55 \times 2^9$

Statement - 2: $S_1=90 \times 2^8$ and $S_2=10 \times 2^8$, then

If $f(x)$ is a function satisfying $f(x+y)=f(x) f(y), \forall x, y \in N$ such that $f(1)=3$ and $\sum_{x=1}^n f(x)=120$. Then, the value of $n$ is

If $S_n=\sum_{r=0}^n \frac{1}{{ }^n C_r}$ and $t_n=\sum_{r=0}^v \frac{r}{{ }^r C_r}$, then $\frac{t_n}{S_s}$ is equal to:

The sum of first $n$ terms of $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\frac{31}{32}+\ldots$ is

The minimum value of $f(x)=\frac{(x-2)^{10}+(x-2)^{11}+3(x-2)^{12}+(x-2)^{15}+(x-2)^{23}+(x-2)^{25}}{(x-2)^{15}}$ $\forall \, x > 2$

Compute the following

(i) $\sqrt[3]{5^{\frac{1}{\log _7 5}}+\frac{1}{\left(-\log _{10} 0.1\right)}}$

(ii) $\log _{0.75} \log _2 \sqrt{\sqrt{\frac{1}{0.125}}}$

(iii) $\left(\frac{1}{49}\right)^{1+\log _7 2}+5^{-\log _{1 / 5} 7}$

(iv) $7^{\log _3 5}+3^{\log _5 7}-5^{\log _3 7}-7^{\log _5 3}$

If $S_n$ denotes the sum of $n$ terms of an AP, then $S_{n+3}-3 S_{n+2}+3 S_{n+1}-S_n$ is equal to

The maximum value of $x^2 y$, subject to constraints $x+y+\sqrt{2 x^2+2 x y+3 y^2}=k$ (constant), $x, y \geq 0$ is

Let the sum of an infinite G.P., whose first term is a and the common ratio is $r$, be 5 . Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first 21 terms of an AP, whose first term is $10 \mathrm{ar}, n^{\text {th }}$ term is $a_n$ and the common difference is $10 ar^2$, is equal to

The sum of the series $04+0.004+000004+\ldots$ is

If $H_1, H_2, H_3, \ldots, H_{2 n+1}$ are in $\mathrm{HP}$, then $\sum_{i=1}^{2 n}(-1)^i\left(\frac{H_i+H_{i+1}}{H_i-H_{i+1}}\right)$ is equal to