General Mathematics · Past QCAA questions All simple familiar questions
2024 · Paper 1 · Simple familiar

Question 16

[3 marks] Scientific calculator
Topic guidesPremiumTopic guide 5 · Recursion, tables and graphs
Game plansPremiumGame plan 3 · Growth and decay in sequences
The question

The number of seats in each row of a theatre forms the terms of the arithmetic sequence $t_{n+1}=t_n+8$, where $t_1=25$.

a)

How many seats are in the second row of the theatre?

[1 mark]
b)

Complete the table and then calculate the total number of seats in the first four rows of the theatre.

[2 marks]
Row 1 2 3 4
Number of seats
Watch the situation first

Eight more seats every row, then add up all four rows

Watch the theatre fill from the stage back. Each row copies the row in front of it and adds 8 seats, shown in blue. Then the four rows go into the table and get added.

Row 1 has 25 seats, and each row adds 8 Write each row into the table Part b) wants all four rows added Pair the first row with the last as a check stage row 1row 2row 3row 4 25 33 41 49 t1 = 25 t2 = 25 + 8 = 33 t3 = 33 + 8 = 41 t4 = 41 + 8 = 49 RowNumber of seats 1234 25 33 41 49 25 + 33 + 41 + 49 = 148 seats 49 is row 4 on its own. The total counts every seat in all four rows. 25 + 49 = 74 and 33 + 41 = 74, so 2 × 74 = 148

An arithmetic sequence adds the same amount each time, so every row is 8 seats longer than the one in front. Part a) stops at row 2. Part b) adds all four rows, which is why the total is far bigger than row 4’s 49.

Now for the mathematics

Start the table at row 1, then choose what to add

Make the three decisions the question needs. The theatre and the table rebuild from your choices, and the feedback names the slip if one creeps in.

Row 1 holds
To get the next row
Part b) total number of seats
stage Row 1Row 2Row 3Row 4 »»»»
Part a) the second row
Your table
Row 1 2 3 4
Number of seats
Your total

Make all three choices, and read the stem for the first two. $t_1=25$ tells you where to start, and $t_{n+1}=t_n+8$ tells you how to step.

$t_1=25$ is row 1 itself. Starting at 33 treats 25 as a row in front of row 1, so every row slides one place back and your total comes out at . Put 25 under Row 1, then step forward from there.

Multiplying by 8 would make a geometric sequence, and row 4 would hold seats. The rule $t_{n+1}=t_n+8$ adds 8 to the row in front, which is what the word arithmetic in the stem tells you too.

Your table is right, but 49 is the number of seats in row 4 alone. The question asks for the total in the first four rows, so add all four numbers in the table.

That’s it. Row 2 has 33 seats, the table reads 25, 33, 41 and 49, and the first four rows hold seats between them.

QCAA marking guide · 3 marks

Work through the solution one mark at a time

Try each step yourself before you reveal it. Part a) is one mark. Part b) has a mark for the table and one for the total.

Step 1 · part a)1 mark

The rule says each row is the row in front plus 8. Put $n=1$ into it to get row 2 from row 1.

$$t_2=t_1+8=25+8=33$$

The second row of the theatre has 33 seats. The formula book’s rule for the $n$th term gives the same answer: $t_2=t_1+(2-1)d=25+1\times8=33$.

Marker: correctly determines the number of seats in the second row

Step 2 · part b)1 mark

Keep stepping the rule, one line per row, and fill the table with all four terms.

$$t_3=t_2+8=33+8=41,\qquad t_4=t_3+8=41+8=49$$

Row 1 2 3 4
Number of seats 25 33 41 49

Row 1 is $t_1=25$ straight from the stem. As a check on the last cell, $t_4=25+(4-1)\times8=49$.

Marker: correctly completes the table to display the first four terms

Step 3 · part b)1 mark

Add every number in the table.

$$25+33+41+49=148$$

The first four rows hold 148 seats

Follow-through applies, so if a slip crept into your table, adding your own four numbers correctly still earns this mark. A quick check: the first and last rows make $25+49=74$, the middle two make $33+41=74$, and $2\times74=148$.

Marker: calculates total number of seats in first four rows

Putting it all together

Each phrase of the question gave you something

“The number of seats in each row of a theatre”
Each term is one row, and $n$ is the row number.
“forms the terms of the arithmetic sequence”
The same amount is added every time. Nothing is multiplied.
“$t_{n+1}=t_n+8$”
Each row has 8 more seats than the row in front, so $d=8$.
“where $t_1=25$”
Row 1 has 25 seats. It goes straight into the first cell of the table.
“How many seats are in the second row”
One step of the rule: $t_2=25+8=33$.
“Complete the table”
Four terms, 25, 33, 41 and 49. The table is its own mark.
“the total number of seats in the first four rows”
Add all four rows to get 148. Row 4 on its own is only 49.
What makes this simple familiar

You are handed the rule and the first term, and the table tells you exactly how many terms to find. The 2024 subject report put generating values from a given arithmetic or geometric sequence on its list of things students did well. Recurrence rules like $t_{n+1}=t_n+8$ are still in the 2025 syllabus, alongside the formula book’s $t_n=t_1+(n-1)d$.

If I had to guess where you might lose marks, it would be the last line of part b). After filling in the table it is easy to write 49, the seats in row 4, as the total. Another easy slip is starting the table at 33 by treating 25 as a row in front of row 1. Write 25 under Row 1 before anything else, add all four cells, and finish with a sentence that says how many seats the first four rows hold.

Only mark this done when you could do it without help. Reading the solution does not count.

Back to the index

Question wording and marking-guide steps are from the 2024 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.

Spot a mistake on this page? Report it