Mathematical Induction (HSC Maths Extension 1, 2026) quiz

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: