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Discussion –

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Discussion –

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# STPM 2016 Mathematics (M) Term 1 Assignment  ## Introduction

The binomial theorem is a quick way of expanding a binomial expression that is raised to some power. When the binomial expressions are expanded, is there any type of pattern developing which might help us expand more quickly? It is fun to investigate how it works out in a pattern.

## Sample Question

Question 1

(a) Expand $(a+ b)^n$  for $n=0, 1, 2, \ldots, 6$.
(b) What observations can we make about the terms in the expansion of $(a + b)^n$?
(c) Write down the general formula for the binomial expansion of $(a + b)^n$.
(d) Obtain the general formula for binomial expansion of $(1 + x)^n$ where $n$ is a rational number, and state the condition for the validity of the binomial expansion with a negative integral exponent or a rational exponent.

Question 2

Deduce the value of

(a) ${n \choose 0}+{n \choose 1}+{n \choose 2}+\ldots+{n \choose n}$

(b) ${n \choose 0}-{n \choose 1}+{n \choose 2}-\ldots+(-1)^n{n \choose n}$

Question 3

The number $e$ may be defined by $e=\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$. Determine the approximate value for $e$.

##### Sample Solution
Scroll down for solution 🙂

## Question 1 (b)

• There are $n+1$ terms in the expansion.
• The first term’s powers start at $n$ and go down.
• The second term’s powers start at 0 and go up.
• The coefficients are from Pascal’s triangle or ${n \choose r}=\frac{n!}{(n-r)!r!}$.

## Question 1 (c)

The general term of the expansion $(a+b)^n$ $(r+1)$-th term is ${n \choose r}a^{n-r} b^r$.

## Question 1 (d)

In short, $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\ldots+\frac{n(n-1)(n-2)(n-3)\ldots(n-r+1)}{r!}x^r+\ldots$. And the expansion is valid for $|x|<1$.

## Question 3

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KK LEE has been a STPM Mathematics tuition teacher since June 2006, but his love of Maths dates back to at least 1999 when he was Form 4. KK LEE started teaching in 2006 at Pusat Tuisyen Kasturi. He was known as "LK" when he was teaching in PTK. After teaching STPM Mathematics for 8 years in PTK, he joined Ai Tuition Centre in 2014. Over the years he has taught Mathematics (T), Mathematics (S), Mathematics (M), Additional Mathematics.

1. THX…

2. Sir, can u explain question 2

• To get the equation, you need to substitute x=1 and x=-1

3. hi, may i know how to write the methodology? i have no idea. QAQ

• For methodology, try to write about the methods/theorem/ways you used to solve the questions.

4. sir, may i know how to write about “communication”? and what is “viva”

• 5. • 