How is Pascal's triangle built?

Every row starts and ends with 1, and each number in between is the sum of the two above it . The first row is a single 1, and each new row is one entry longer than the last.

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Binomial Theorem (Cambridge (CIE) IGCSE Additional Maths): Flashcards

Exam code: 0606

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FrontBinomial Theorem
Complete the start of the binomial theorem, for positive integer $n$ :
$(a + b)^{n} = a^{n} + (\binom{n}{1})______b + (\binom{n}{2}) a^{n - 2}______+ \dots$

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Cards in this collection (9)

Complete the start of the binomial theorem, for positive integer $n$ :
$(a + b)^{n} = a^{n} + (\binom{n}{1})______b + (\binom{n}{2}) a^{n - 2}______+ \dots$
The completed expansion begins:
$(a + b)^{n} = a^{n} + (\binom{n}{1}) a^{n - 1} b + (\binom{n}{2}) a^{n - 2} b^{2} + \dots$
The power of $a$ falls by one each time while the power of $b$ rises by one.
How is Pascal's triangle built?
Every row starts and ends with 1, and each number in between is the sum of the two above it.
The first row is a single 1, and each new row is one entry longer than the last.
How does Pascal's triangle relate to the binomial coefficients?
Row $n$ of the triangle lists the values of $(\binom{n}{r})$ , counting the top row as row 0 and the first entry of each row as $r = 0$ .
It is a quick way to get coefficients for small $n$ without a calculator, but slow and error-prone once $n$ grows.
Complete the general term of the expansion of $(a + b)^{n}$ by filling in the two missing powers:
$(\binom{n}{r}) a^{______} b^{______}$
The completed general term is:
$(\binom{n}{r}) a^{n - r} b^{r}$
This single expression is what you use when a question wants one particular term rather than the whole expansion.
True or False?
In every term of the expansion of $(a + b)^{n}$ , the powers of $a$ and $b$ add up to $n$ .
True.
The general term is $(\binom{n}{r}) a^{n - r} b^{r}$ , and $(n - r) + r = n$ whatever $r$ is.
This makes a quick check on any expansion: if the powers in a term do not total $n$ , something has gone wrong.
To find the coefficient of $x^{r}$ in $(a + b x)^{n}$ , how do you know which value of $r$ to use?
The $x$ sits only in the second bracket term, so the term $(\binom{n}{r}) a^{n - r} (b x)^{r}$ carries exactly $x^{r}$ .
When the bracket contains $x^{2}$ or $\frac{1}{x}$ instead, work out the power of $x$ the general term produces and set it equal to the one you want.
What is the difference between ascending and descending powers of $x$ ?
Ascending starts with the constant term $a^{n}$ and works up through increasing powers of $x$ .
Descending starts at the other end, with the highest power of $x$ , so the same terms are simply written in reverse.
Find the first three terms of $(3 - 2 x)^{5}$ in ascending powers of $x$ .
Take $a = 3$ and $b = - 2 x$ , keeping the negative sign inside the bracket:
$3^{5} + (\binom{5}{1}) 3^{4} (- 2 x) + (\binom{5}{2}) 3^{3} (- 2 x)^{2}$
This simplifies to $243 - 810 x + 1080 x^{2} + \dots$ , where the third term is positive because $(- 2 x)^{2}$ is.
In the expansion of ${(3 x + \frac{2}{x})}^{8}$ , which term is the constant one?
The general term contains $x^{8 - r} \times x^{- r}$ , which is $x^{8 - 2 r}$ .
A constant term needs $8 - 2 r = 0$ , so it is the term with $r = 4$ .

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