Sequence of series practice · Medium

Jairam purchased a house in Rs 15000 and paid Rs 5000 at once. Rest money he promised to pay in annual installment of Rs 1000 with 10% per annum interest. How much money is to be paid by Jairam?

If $\lambda=\sum\limits_{{i=1}}^{\infty} \frac{1}{i^4}$, then $\sum\limits_{{i=1}}^{\infty} \frac{1}{(2 i-1)^4}$ is

If $\frac{1}{2} \operatorname{cosec}^2 \theta, 2 \cot \theta, \sec \theta, 0<\theta<\frac{\pi}{2}$ are in GP, then $\theta$ is equal to

Let $a=2^{\frac{1}{50}}-1$ and for each $n \geq 2, S_n=C_1+C_2 \cdot a+C_3 \cdot a^2+\ldots \ldots+C_n \cdot a^{n-1}$ $\left({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=\mathrm{C}_{\mathrm{r}}\right)$ the $S_{101}-S_{100}=$

If $a, b, c$ are in A.P., then $\frac{1}{\sqrt{a}+\sqrt{b}}, \frac{1}{\sqrt{a}+\sqrt{c}}, \frac{1}{\sqrt{b}+\sqrt{c}}$ are in

The harmonic mean, arithmetic mean and geometric mean of two numbers are $4, \mathrm{A}$ and $\mathrm{G}$ respectively. If $2 \mathrm{A}+\mathrm{G}^2=27$, then the numbers are

If $\log _e 5, \log _e\left(5^x-1\right)$ and $\log _e\left(5^x-\frac{11}{5}\right)$ are in A.P then the values of $x$ are

Minimize $z=\sum_{j=1}^n \sum_{i=1}^m c_{i j} x_{i j}$
Subject To: $\sum_{j=1}^n x_{i j} \leq a_i, i=1, \ldots \ldots \ldots . m$
$\sum_{i=1}^m x_{i j}=b_i, j=1, \ldots \ldots \ldots . n$
is a (L.P.P.) with number of constraints

If four numbers in GP are such that the third number is greater than the first by 9 and the second number is greater than the fourth by 18 , then the numbers are

If $1^3=1,2^3=3+5,3^3=7+9+11$ and $4^3=13+15+17+19$. then $n^3$ as a similar series is

A total of $n$ arithmetic means are inserted between $x$ and $2 y$ and further $n$ arithmetic means are inserted between $2 x$ and $y$. If the $k$ th arithmetic means of both sets are equal, then a relation between $x$ and $y$ is

The value of $\mathrm{l}^{2}+3^{2}+5^{2}+$ $+25^{2}$ is :

The sum of all two digit numbers which when divided by 4 , yield unity as remainder, is

$\mathrm{S}_n$ for the series $\left(1^2-2^2\right)+\left(3^2-4^2\right)+\left(5^2-6^2\right)+\ldots$ is

$2+4+7+11+16+\ldots$ to $n$ terms $=$

Four numbers are in arithmetic progression. The sum of first and last term is 8 and the product of both middle terms is 15 . The least number of the sequence is

If $a,\, b, \, c$ are in A.P; $b, \, c, \, d$ are in G.P; $c, \, d, \, e$ are in H.P, then $\frac{1}{a}, \, \frac{1}{c}, \, \frac{1}{e}$ are in

The value of $0.423,$ is

If the sequence $< a_n >$ is defined by $a_n=3 n+2$, then difference of any two consecutive terms is

The sum of n terms of the series $1 + \left( {1 + x} \right) + \left( {1 + x + {x^2}} \right) + \;...$ will be

If $a^x=b^y=c^z$ and $a, b, c$ are in GP, then $x, y$ and $=$ are in

The sum of first four terms of a GP is $12(1-\sqrt{5})$ If the common ratio is $-\sqrt{5}$, then the first term of a GP is

If $x,y,z$ be three positive prime numbers. The progression in which $\sqrt x ,\sqrt y ,\sqrt z$ can be three terms (not necessarily consecutive) is

The first and last terms of an AP are 1 and 11 . If the sum of its terms is 36 , then the number of terms will be

The sum of series $1 + \frac{3}{2} + \frac{7}{4} + \frac{{15}}{8} + \frac{{31}}{{16}} + \;...$ is equal to

Let $a_{n}$ be $n^{\text {th }}$ term of the series $5+8+14+23+35 +50+\ldots$ and $S_{n}=\sum\limits_{{k=1}}^{n} a_{k}$. Then $S_{30}-a_{40}$ is equal to

The sum of the first 20 terms of the series $5+11+$ $19+29+41+\ldots$ is

If $a$ and $b$ are two positive numbers, then

If the sum of first $n$ terms of a series is $5 n^2+2 n$, then its second term is

If ${\log _2}a + {\log _4}b + {\log _4}c = 2$

${\log _9}a + {\log _3}b + {\log _9}c = 2$

${\log _{16}}a + {\log _{16}}b + {\log _4}c = 2$ then