# Recent content by juantheron

1. ### Irrational Integration

(a)\; $Evaluation of$\int\frac{\sqrt{9x^2+4x+6}}{8x^2+4x+6}dx$(b)\;$Evaluation of $\int\frac{x^2(x\sec x+\tan x)}{(x\tan x-1)^2}dx.$

9. ### Sequence and series

Thanks camelrider.
10. ### Sequence and series

$Let$\displaystyle a_{n} = n+\frac{1}{n}$for$n=1,2,3,4,....20$and$p=\frac{1}{20}\sum^{20}_{n=1}a_{n}And $q = \frac{1}{20}\sum^{20}_{n=1}\frac{1}{a_{n}}$. Then proving $q\in \left(0,\frac{21-p}{21}\right)$. Trial p = \frac{1}{20}\left((1+2+3+\cdot +20)+1+\frac{1}{2}+\frac{1}{3}+\cdots...
11. ### Largest prime number

$Finding largest prime number which divide$\displaystyle \binom{2000}{1000}We can write $\displaystyle \binom{2000}{1000} = \frac{2000!}{1000!\cdot 1000!} = \frac{1001 \cdot 1002 \cdot 1003\cdots \cdots 2000}{1000!}$ i did not understand how to solve further, please explain me

14. ### problem involving floor sum

\displaystyle \bigg\lfloor\frac{2017!}{1!+2!+3!+\cdots \cdots +2016!}\bigg\rfloor $where$\lfloor x \rfloor $is a floor of$xFor upper bound $1!+2!+3!+\cdots \cdots +2016!=2016(1!+2!+3!+\cdots +2016!)$ 2016(1!+2!+3!+\cdots \cdots +2016!)>2016(2016!+2015!)=2017! \displaystyle...