State the two main steps required in a proof by mathematical induction.

For the statement \(1+2+\cdots+n=\frac{n(n+1)}2\) for \(n\ge1\), verify the base case.

Write the inductive hypothesis for proving \(1+3+5+\cdots+(2n-1)=n^2\) for \(n\ge1\).

If \(P(k)\) is \(2+4+\cdots+2k=k(k+1)\), write the statement \(P(k+1)\).

Complete the inductive step: if \(1+2+\cdots+k=\frac{k(k+1)}2\), show \(1+2+\cdots+k+(k+1)=\frac{(k+1)(k+2)}2\).

Verify the base case for \(2^n\ge n+1\) for all integers \(n\ge1\).

In an induction proof, why must \(k\) be arbitrary rather than a specific value such as \(k=5\)?

For \(P(n): 3^n-1\) is divisible by \(2\), write the inductive hypothesis and the target for \(P(k+1)\).

Prove by induction that \(2+4+\cdots+2n=n(n+1)\) for all \(n\ge1\).

Prove by induction that \(1+3+5+\cdots+(2n-1)=n^2\) for all \(n\ge1\).

Explain why the base case cannot be omitted from an induction proof.

Find the flaw: A proof assumes \(P(k+1)\) is true and then derives \(P(k)\). Why is this not a standard induction proof of all cases from \(1\) upward?

Prove by induction that \(\sum_{i=1}^{n} i^2=\frac{n(n+1)(2n+1)}6\) for all \(n\ge1\).

Prove by induction that \(3^n-1\) is divisible by \(2\) for all \(n\ge1\).

Prove by induction that \(5^n-1\) is divisible by \(4\) for all \(n\ge1\).

Prove by induction that \(2^n\ge n+1\) for all \(n\ge1\).

A statement is claimed for all integers \(n\ge4\). What base case should be checked, and how does the induction step change?

Diagnose the error: "The formula is true for \(n=1\), \(2\), and \(3\), so it is true for all \(n\)."

Prove by induction that \(7^n-2^n\) is divisible by \(5\) for all \(n\ge1\).

A proposed induction proof of \(1+2+\cdots+n=n^2\) uses a valid-looking inductive step. Explain why the claim is still false.