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STPM PBS Assignment

STPM Mathematics (M) Assignment

By KK LEE

August 4, 2015

STPM 2016 MM Term 1 Assignment Coursework answer

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 (a)

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 2

Question 3

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