2022 HCI J1 BT Q7
It is given that ${{u}_{r}}=\frac{1}{\left( r+2k \right)!}$, where $r\in {{\mathbb{Z}}^{+}}$ and $k$ is a positive constant.
(i)
Show that ${{u}_{r}}-{{u}_{r+1}}=\frac{r+2k}{\left( r+2k+1 \right)!}$.
[1]
(ii)
Hence use the method of differences to find $\sum\limits_{r=1}^{n}{\frac{r+2k}{\left( r+2k+1 \right)!}}$ in terms of $n$ and $k$. You need not simplify your answer.
[3]
Let ${{S}_{n}}$ denote the sum of the first $n$ terms of the series
$\frac{1}{4\,\,\,2!}+\frac{1}{5\,\,\,3!}+\frac{1}{6\,\,\,4!}+…$.
(iii)
By finding a suitable integer value of k and using the result obtained in part (ii), find ${{S}_{n}}$ in terms of $n$ .
[2]
(iv)
Deduce that ${{S}_{n}}<\frac{1}{6}$ for all $n\in {{\mathbb{Z}}^{+}}$.
[1]
(v)
Find the least value of $n$ for which ${{S}_{n}}$ is within ${{10}^{-20}}$ of the sum to infinity.
[2]
Suggested Video
Share with your friends!
- No charge
- Unsubscribe anytime
Continue reading
Introducing Our Sec 1 Assessment Book: From Concepts to Confidence
Welcome to Tim Gan Math Learning Centre, where we have been dedicated to nurturing mathematical excellence since our establishment in 2017. At our centre, we are committed to providing quality
Entangled in the Web: Managing Online Entertainment, Internet Addiction, and Academic Achievement in Math
In today’s digital age, the internet has seamlessly woven itself into the fabric of our daily lives, becoming an indispensable tool for communication, information, and, notably, entertainment. The allure of
Goal Setting and Progress Tracking for Math Students: A Comprehensive Guide
In the pursuit of academic success, setting and achieving goals play a pivotal role, particularly in disciplines like mathematics. Effective goal setting not only provides students with a roadmap for