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(a) This is Row 3 in a house of cards. Complete these statements for Row 3.
There are ............... horizontal cards.
There are ............... diagonal cards.
There are ............... cards in total. [1]
(b) (i) On the grid, draw Row 4 in a house of cards. [1]
(ii) Complete these statements for Row 4.
There are ............... horizontal cards.
There are ............... diagonal cards.
There are ............... cards in total. [1]
(c) Complete the table. Use your answers to part (a), part (b)(ii) and any patterns you notice.
[Table_1]
[2]
(d) Find an expression, in terms of $n$, for
(i) the number of horizontal cards in Row $n$ ............................................... [1]
(ii) the number of diagonal cards in Row $n$ ............................................... [1]
(iii) the total number of cards in Row $n$. Give your answer in its simplest form. ............................................... [2]
(e) Work out the number of horizontal cards, the number of diagonal cards and the total number of cards in Row 40.
Number of horizontal cards ...............................................
Number of diagonal cards ...............................................
Total number of cards ............................................... [3]
(f) The total number of cards in Row $p$ is 368. Work out how many diagonal cards are in Row $p$. ............................................... [4]
500
589
543
(a) This is House 3.
Complete these statements for House 3.
There are ................ horizontal cards.
There are ................ diagonal cards.
There are ................ cards in total.
(b) On this grid, draw House 4.
(c) Complete the table.
Use your answers to part (a), part (b) and any patterns you notice.
[Table_1]
(d) Write down the rule to continue the sequence for the number of horizontal cards.
.........................................................................................................................................................
(e) This is an expression for the number of horizontal cards in House $h$.
$0.5h(h-1)$
Show that this expression gives the same result as your rule in part (d) when $h = 6$.
(f) Find an expression, in terms of $h$, for the number of diagonal cards in House $h$.
..................................................
(g) Use the expression from part (e) and your answer from part (f) to find an expression for the total number of cards in House $h$.
Give your answer in its simplest form.
..................................................
(h) Lee makes a house with 100 cards.
Find the number of cards in the bottom row.
..................................................
509
558
572
