Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: From Patterns to Generalizations: Sequences and Series, Exercise 47: Exercise 1O

The values in the fourth row of Pascal's triangle are $1,3,3,1$.

Write down the values in the next row of the triangle.

The values in the fourth row of Pascal's triangle are $1,3,3,1$.

Find the term in $x$ in ${\left(3x-2\right)}^4$.

Expand ${\left(x-5\right)}^5$ using the binomial theorem.

The term in $x^4$ in $\left(2x-1\right){\left(x-5\right)}^5$ is $k{x}^4$, then find the value of $k$.

Show that the expansions of $(3-2x{)}^4$ and $(-2x+3{)}^4$ are the same.

Will $(b-ax{)}^n$ and $(-ax+b{)}^n$ always have the same expansion? Explain your answer.

Find the binomial power, in the form ${\left(a+b\right)}^n$, with expansion $27x^3-108x^{2}y+144x{y}^2-64y^3$.

Show that ${}^{n}C_0+{}^{n}C_1+{}^{n}C_2+\dots ..+{}^{n}C_n=2^n$.