Embibe Experts Solutions for Chapter: Sequence & Series, Exercise 2: Exercise

The line $x+y=1$ meets $x$-axis at $A$ and $y$-axis at $B,$ $P$ is the mid-point of $AB\mathit{;}\mathit{ }{P}_1$  is the foot of the perpendicular from $P$ to $OA; M_1$ is that of $P_1$ on $OP; P_2$ is that of $M_1$ on $OA; M_2$ is that of $P_2$ on $OP; P_3$ is that of $M_2$ on $OA;$ and so on.

If $P_n$ denotes the ${\mathrm{n}}^{\mathrm{th}}$ foot of the perpendicular on $OA;$ then $O{P}_n$ is equal to

Suppose that $F\left(n+1\right)=\frac{2F\left(n\right)+1}{2}$ for $n=1, 2, 3, \dots$ and $F\left(1\right)=2.$ Then $F\left(101\right)$ equals

Let $a_1, a_2, a_3, \dots$ be in $\mathrm{A}.\mathrm{P}.$ and $g_1, g_2, \dots$ be in $\mathrm{G}.\mathrm{P}.$. If $a_1=g_1=2$ and $a_{10}=g_{10}=3,$ then

Statement-1: $(666\dots \dots ndigits{)}^2+\left(888\dots \dots n \mathrm{digits}\right)=\left(444......2n \mathrm{digits}\right)$

Statement-2: $\left(\underset{12\text{ times }}{\underbrace{111\dots \dots .1}}\right)$ is a prime number

In a geometric progression with common ratio $\text{'}q\text{'}$ the sum of the first $109$ terms exceeds the sum of the first $100$ terms by $12.$ If the sum of the first nine terms of the progression is $\frac{\lambda }{q^{100}},$ then the value of $\lambda$ equals to

If ${\alpha }_1, {\alpha }_2, \dots \dots {\alpha }_6\left({\alpha }_i>0\right)$ are the roots of the equation $x^6-12x^5+a{x}^4-b{x}^3+c{x}^2-dx+64=0$, then $b$ is equal to

$\sum _{k=1}^n\frac{k}{k^4+\frac{1}{4}}=$

If $\sin \alpha , \sin \beta , \sin \gamma$ are in $\mathrm{A}.\mathrm{P}.$ and $\cos \alpha , \cos \beta , \cos \gamma$ are in $\mathrm{G}.\mathrm{P}.,$ then $\frac{{\cos }^2\alpha +{\cos }^2\gamma -4\cos \alpha \cos \gamma }{1-\sin \alpha \sin \gamma }=$