Expand $(x-1{)}^4$ completely.

Which list gives the coefficients of the expansion of $(a+b{)}^6$ in order?

In the expansion of $(x-3{)}^7$ written in descending powers of $x$, what is the sixth term?

In a proof by induction that $n!>2^n$ for every integer $n\ge 4$, what is assumed at the start of the inductive step?

In a proof by induction that $2+4+6+\cdots +2n=n(n+1)$ for every integer $n\ge 1$, what is assumed at the start of the inductive step?

The repeating decimal $0.\overline{12}$ is the sum of an infinite geometric series whose first term is $\frac{12}{100}$. What fraction does it equal, in lowest terms?

What is the coefficient of $x^2$ in the expansion of $(2x+2{)}^5$?

The repeating decimal $0.\overline{81}$ is the sum of an infinite geometric series whose first term is $\frac{81}{100}$. What fraction does it equal, in lowest terms?

Evaluate ${\sum }_{n=1}^{\infty }15{\left(-\frac{1}{10}\right)}^{n-1}$, giving an exact fraction.

Evaluate ${\sum }_{n=1}^{\infty }7{\left(\frac{5}{8}\right)}^{n-1}$, giving an exact fraction.

An infinite geometric series has first term $20$ and sum $50$. What is its common ratio?

Find the sum of the infinite series $9-\frac{27}{4}+\frac{81}{16}-\frac{243}{64}+\cdots$.

In the expansion of $(x-3{)}^6$, what is the term containing $x^2$?

How many terms does the expansion of $(a+b{)}^{11}$ have once it is written out?

In the expansion of $(a+b{)}^9$ written in descending powers of $a$, what is the last term?

In the expansion of $(a+b{)}^8$ written in descending powers of $a$, what is the middle term?

Find the sum of the infinite series $4-3+\frac{9}{4}-\frac{27}{16}+\cdots$.

A proof by induction of $1+3+5+\cdots +(2n-1)=n^2$ has assumed the claim at $n=k$. What must the inductive step arrive at?

Find the sum of the infinite series $6-\frac{21}{4}+\frac{147}{32}-\frac{1029}{256}+\cdots$.

A proof by induction claims that $2+4+6+\cdots +2n=n(n+1)$ for every integer $n\ge 1$. What does its base case verify?