smc-7422-30-Other Proofs

For integers `n ≥ 1`, the triangular numbers `t_n` are defined by `t_n = (n(n + 1))/2`, giving `t_1 = 1, t_2 = 3, t_3 = 6, t_4 = 10` and so on.

For integers `n >= 1`, the hexagonal numbers `h_n` are defined by `h_n = 2n^2-n`, giving `h_1 = 1, h_2 = 6, h_3 = 15, h_4 = 28` and so on.

Show that the triangular numbers `t_1, t_3 , t_5`, and so on, are also hexagonal numbers. (2 marks)

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Show that the triangular numbers `t_2, t_4 , t_6`, and so on, are not hexagonal numbers. (1 mark)

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i. `text(See Worked Solution)`

ii. `text(See Worked Solution)`

Show Worked Solution

i. `t_n = (n(n + 1))/2`

♦ Mean mark (i) 50%.

`text(Odd triangular numbers:)`

`→ \ text(Let)\ \ n = 2k-1\ \ text(for integers)\ \ k >= 1`

`t_(2k-1)`	`= ((2k-1)(2k-1 + 1))/2`
	`= (2k(2k-1))/2`
	`= 2k^2-k\ \ text{(definition of hexagonal numbers)}`

`:. t_1, t_3\ …\ text(are also hexagonal numbers.)`

ii. `t_2, t_4, t_6, …`

♦ Mean mark (ii) 1%!

`→\ text(Let)\ \ n = 2k\ \ text(for integers)\ \ k > = 1`

`t_(2k)`	`= (2k(2k + 1))/2`
	`= 2k^2 + k`

`text(Find values for)\ \ k, n\ \ text(that satisfy:)`

`k = -n =>\ text(no solution)\ (k, n >= 1)`

`2k = 2n-1 =>\ text(no solution)\ \ (2k\ \ text(is even,)\ \ 2n-1\ \ text{is odd)}`

`:. t_2, t_4, t_6, …\ text(are not hexagonal numbers.)`

Four cards are placed on a table with a letter on one face and a shape on the other.

You are given the rule: "if N is on a card then a circle is on the other side."

Which cards need to be turned over to check if this rule holds?

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`=> \ D`

Show Worked Solution

`text(Solution 1)`

`text(L)text(ogically equivalent statements are:)`

`N`	`=> \ text(circle)`
`not\ text(circle)`	`=> not N`

`text(To confirm rule is not broken,)`

`N \ text(must be turned)`

`text{Triangle (not circle) must be turned – only other shape not a circle.}`

`=> D`

`text(Solution 2)`

`text(Consider the flip side of each card.)`

`text(If circle has) \ N \ text(on the other side (or not)) – text(tells us nothing.)`

`text(If) \ G \ text(has a circle on the other side (or not)) – text(tells us nothing.)`

`text(If) \ N \ text(doesn’t have a circle on other side) – text(rule broken.)`

`text(If triangle has an) \ N \ text(on other side) – text(rule broken.)`

`:. \ text(Need to turn) \ N \ text(and triangle)`

Proof, EXT2 P1 2021 HSC 15b