Proof

Section CP-1

1 spec leaf

Notes and three levels of exam-style practice for each registered specification leaf in this section.

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CP-1.1

Construct proofs using mathematical induction. Contexts include sums of series, divisibility and powers of matrices.

Mathematical induction proves a statement $P(n)$ for every integer from a stated starting value: verify the base case, assume $P(k)$ , prove $P(k+1)$ , then conclude.
In the inductive step, use the hypothesis explicitly: add the next term for a sum, isolate a multiple for divisibility, or multiply by the matrix once more for a matrix power.
For example, after assuming a sum formula for $S_k$ , write $S_{k+1}=S_k+$ the term with index $k+1$ , then simplify to the target formula with $k+1$ substituted for $n$ .
A common error is circular reasoning: assuming the result for $k+1$ , or checking several numerical cases, does not prove the result for all permitted integers.

Tier 1 · Easy

4 marks

ORIGINAL

Prove by mathematical induction that $1+3+5+\cdots+(2n-1)=n^2$ for every positive integer $n$ .

Tier 2 · Standard

5 marks

ORIGINAL

Prove by mathematical induction that $8^n-1$ is divisible by $7$ for every positive integer $n$ .

Tier 3 · Hard

6 marks

ORIGINAL

Let $A=\begin{pmatrix}2&1\\0&2\end{pmatrix}$ . Prove by mathematical induction that $A^n=\begin{pmatrix}2^n&n2^{n-1}\\0&2^n\end{pmatrix}$ for every positive integer $n$ .