Which four parts make up the standard structure of a proof by mathematical induction?

Base case, inductive hypothesis, inductive step, conclusion

In a proof that the statement $P(n)$ holds, what does the inductive hypothesis assume?

That $P(k)$ is true for some integer $k$

Proving $2^n > n^2$ for $n \ge 5$ by induction, why must the base case be $n = 5$ rather than $n = 1$?

The inequality $2^n > n^2$ fails for some smaller $n$ such as $n = 2, 3, 4$

§-Quiz

Mathematical Induction (HSC Maths Extension 1, 2026) quiz

10 questions. Pick an answer and you'll see why right away. Options are shuffled each attempt, and the review at the end links each missed question to the dot point to revise.

Which four parts make up the standard structure of a proof by mathematical induction?
In a proof that the statement $P(n)$ holds, what does the inductive hypothesis assume?
Proving $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$ , what is the base case value at $n = 1$ for both sides?
In proving $\sum_{i=1}^{n} (2i-1) = n^2$ , the $(k+1)$ -th term added to the sum is:
Proving $7^n - 1$ is divisible by $6$ , the hypothesis gives $7^k = 6M + 1$ . What is $7^{k+1} - 1$ in factored form?
Proving $5^n + 3$ is divisible by $4$ , what is the base case $n = 1$ value of $5^n + 3$ ?
Proving $2^n > n^2$ for $n \ge 5$ by induction, why must the base case be $n = 5$ rather than $n = 1$ ?
Proving $9^n - 1$ is divisible by $8$ , the hypothesis is $9^k = 8M + 1$ . What is $9^{k+1} - 1$ in factored form?
Proving $\sum_{i=1}^{n} 2^i = 2^{n+1} - 2$ , adding the term $2^{k+1}$ to $(2^{k+1} - 2)$ gives:
For the sequence $a_1 = 1$ , $a_{n+1} = 3 a_n + 2$ , applying the recursion to the hypothesis $a_k = 2 \cdot 3^{k-1} - 1$ gives $a_{k+1}$ equal to: