$$(a + b)^12$$ What would you use to expand this??
The binomial theorem.
What is the $r$-th term in the binomial expansion of $(a + b)^n$??
$$ \left(\begin{matrix} n \\ r \end{matrix}\right) a^n b^{n - r} $$
$$1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1$$ What is the next row of Pascal’s triangle??
$$ 1 \quad 6 \quad 15 \quad 20 \quad 15 \quad 6 \quad 1 $$
What does the $n$-th row of Pascal’s triangle start with (ignoring the top)??
$$ 1 \quad n $$
What are the coefficients for $(a + b)^3$??
$$ 1 \quad 3 \quad 3 \quad 1 $$
What is $(a + b)^3$??
$$ a^3 + 3a^2b + 3ab^2 + b^3 $$
$$ \left(\begin{matrix} 8 \ 3 \end{matrix}\right) $$ Because of the symmetry property, what is this equal to??
$$ \left(\begin{matrix} 8 \\ 5 \end{matrix}\right) $$
$$ \left(\begin{matrix} n \ r \end{matrix}\right) $$ Because of the symmetry property, what is this equal to??
$$ \left(\begin{matrix} n \\ n-r \end{matrix}\right) $$
What would be the expression for working out the $x^3$ term of $(2x + 6)^7$??
$$ \left(\begin{matrix} 7 \\ 3 \end{matrix}\right) (2x)^3 (6)^4 $$
2021-10-14
What is the formula for $(1 + x)^n$ where $|x| < 1$??
$$ 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + … + \frac{n(n-1)(n-2)…(n-(r-1))}{r!}x^r $$
Why does the result $$(1 + x)^n \equiv 1 + nx + \frac{n(n-1)}{2!}x^2 + …$$ hold when $n > 1$ even though the sequence is infinite??
Because you get $0$ in the numerator for later terms and so they disappear.
What is the $x^2$ term in the formula for $(1 + x)^n$??
$$ \frac{n(n-1)}{2!} x^2 $$
What is the $x^7$ term in the formula for $(1 + x)^n$??
$$ \frac{n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)}{7!} x^7 $$
When is the expansion for $$(1 + x)^n$$ valid??
$$ |x| < 1 $$
When is the expansion for $$(1 + 4x)^n$$ valid??
$$ |x| < \frac{1}{4} $$
When is the approximation for $$(1 + x)^n$$ the best??
When the values of $x$ are small.
2021-10-22
What is $$(a + bx)^n$$ equivalent too??
$$ a^n\left( 1 + \frac{b}{a}x \right)^n $$
How would you tackle finding the binomial expansion for $$\frac{4 - 5x}{(1 + x)(2 - x)}$$??
Use partial fractions.
2022-01-11
How would you tackle finding the binomial expansion for $$\sqrt{\frac{1-x}{1+4x}}$$??
Treat it as $(1-x)^{\frac{1}{2}}(1+4x)^{-\frac{1}{2}}$ and multiply the expansions together.
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date: 2021-03-01 11:28
tags:
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- '@?maths'
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title: Maths - Binomial Theorem