Proof by induction：練習問題

When proving by induction that for all positive integers \(n\), the sum of the series \(\sum_{r=1}^{n} (3r-1)\) is given by \(\frac{n(3n+1)}{2}\), what is the sum for the base case \(n=1\)?

Prove by induction that $$9^n - 1$$ is divisible by 8 for all positive integers $n$. When trying to show the result holds for $n=k+1$ assuming it holds for $n=k$, which expression correctly represents $$9^{k+1} - 1$$ in terms of $$(9^k - 1)$$?

Prove by induction that for all integers \(n \ge 1\), \(3^{2n+1} + 2^{n+2}\) is divisible by 7. In the inductive step, assuming the statement is true for \(n=k\), which of the following expressions correctly represents \(3^{2k+3} + 2^{k+3}\) in a form that demonstrates divisibility by 7?

Prove by induction that for all positive integers $$n$$, the sum of the series $$\sum_{r=1}^{n} r(r+2)$$ is given by $$\frac{n(n+1)(2n+7)}{6}$$. Assuming the formula holds for $$n=k$$, which expression correctly represents the sum for $$n=k+1$$?

Prove by induction that $$7^n - 1$$ is divisible by 6 for all positive integers $$n \ge 1$$. Which of the following is a key step in showing that if $$7^k - 1$$ is divisible by 6, then $$7^{k+1} - 1$$ is also divisible by 6?

Prove by induction that for all positive integers $n$, $$\sum_{r=1}^{n} (2r-1) = n^2.$$

Prove by induction that $$7^n - 4^n$$ is divisible by 3 for all positive integers $$n$$.

Prove by induction that for all positive integers \( n \), \( n^3 + 5n \) is divisible by 6.

Prove by induction that $$3^{2n} + 7$$ is divisible by 8 for all positive integers $$n$$.