A proof by induction claims that $3^n>n^3$ for every integer $n\ge 4$. What does the base case verify?

A proof by induction claims that $1+3+5+\cdots +(2n-1)=n^2$ for every integer $n\ge 3$. Which equation belongs in the base case?

For the claim $5\mid (n^5-n)$ for every integer $n\ge 2$, what must the base case establish?

A proof claims that $1+2+2^2+\cdots +2^n=2^{n+1}-1$ for every integer $n\ge 0$. What is its base case?

A proof by induction claims that $2^n\ge n+1$ for every integer $n\ge 3$. What does the required base case check?

In an induction proof of $1+2+\cdots +n=\frac{n(n+1)}{2}$ for every integer $n\ge 1$, what is the inductive hypothesis?

For a proof that $2+4+6+\cdots +2n=n(n+1)$ for every integer $n\ge 1$, which statement is assumed before proving the next case?

A proof claims that $4^n-1$ is divisible by 3 for every integer $n\ge 2$. What is the proper inductive hypothesis?

In a proof of $1^2+2^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}$ for every integer $n\ge 1$, what may be used as the inductive hypothesis?

A proof seeks to show $7^n-1$ is divisible by 6 for every integer $n\ge 1$. Which assumption is valid for its inductive step?

An induction proof of $1+2+\cdots +n=\frac{n(n+1)}{2}$ assumes the formula for $n=k$. What must the inductive step prove?

A proof by induction assumes $2+4+\cdots +2k=k(k+1)$. Which statement must it reach to complete the inductive step?

For the claim $1^3+2^3+\cdots +n^3={\left(\frac{n(n+1)}{2}\right)}^2$, the formula is assumed at $n=k$. What is the target statement?

A proof claims $3^n\ge 2n+1$ for every integer $n\ge 1$ and assumes the claim when $n=k$. Which inequality must be shown next?

An induction proof concerns $1+3+5+\cdots +(2n-1)=n^2$ for every integer $n\ge 1$. If the formula is assumed for $n=k$, what must be proved?

A proof assumes $1+2+\cdots +k=\frac{k(k+1)}{2}$. Which rewrite correctly begins the case $n=k+1$?

For $2+4+6+\cdots +2n=n(n+1)$, suppose the formula has been assumed at $n=k$. How should the left side for $n=k+1$ be rewritten using that assumption?

In proving $1^2+2^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}$, the formula is assumed at $n=k$. Which rewrite uses the hypothesis in the next case?

A student verifies a claim at $n=1$, $n=2$, and $n=3$, then concludes it is true for every integer $n\ge 1$. What essential part of an induction proof is missing?

A student states an inductive hypothesis for a formula at $n=k$ and derives the formula at $n=k+1$, but never checks any initial value in the stated range $n\ge 5$. What is missing?