Algebraic Proof Flashcards

All 6 cards in this deck

What are the steps to prove an expression is always positive by completing the square?

e.g. $n^2 + 6n + 11$

1. Complete the square: $(n+3)^2 + 2$
2. State $(n+3)^2 \ge 0$ for all $n$
3. Conclude: expression $\ge 2$, so it is always positive (least value 2).

True or false? Checking that a statement works for $n = 1$, $2$ and $3$ proves it is true for every integer $n$.

False. Testing values is not a proof — you need a general algebraic argument; but a single counter-example is enough to disprove a statement.

In an algebraic proof, what expression represents any odd number (where $n$ is an integer)?

$2n+1$ (or $2n-1$).
Any even number is $2n$.

What are the steps to prove a difference of two squares is divisible by a given number?
e.g. prove $(n+5)^2-(n+3)^2$ is divisible by 4 for any integer $n$

1. Expand both squares: $n^2+10n+25$ and $n^2+6n+9$
2. Subtract and simplify: $4n+16$
3. Factorise and conclude: $4(n+4)$, which is divisible by 4

A sequence has $n$th term $X_n = n^2-1$. What expression gives the term straight after $X_n$?

$X_{n+1}=(n+1)^2-1$.
Replace every $n$ in the $n$th term by $n+1$.

True or false? Every term of the sequence with $n$th term $4n+2$ is shown to be even by writing it as $2(2n+1)$.

True.
It is 2 times an integer expression, so every term is even.