A theatre has $18$ seats in the first row. Each later row has $3$ more seats than the row in front of it. Find the number of seats in row $11$, and the total number of seats in the first $11$ rows.
Question HKC78-AASL-SEQ-01
[Maximum mark: 5]
A theatre has $18$ seats in the first row. Each later row has $3$ more seats than the row in front of it. Find the number of seats in row $11$, and the total number of seats in the first $11$ rows.
Mark Scheme
This is arithmetic with $u_{1} = 18$ and $d = 3$.
Row $11$: $u_{11} = 18 + 10 cdot 3 = 48$. [2]
Total seats: $S_{11} = dfrac{11}{2}(18 + 48) = dfrac{11}{2} cdot 66 = 363$. [3]
Total: 5 marks
A geometric sequence has $u_{1} = 8$ and $u_{4} = 27$. The common ratio is positive. Find the common ratio and the sixth term.
Question HKC78-AASL-SEQ-02
[Maximum mark: 5]
A geometric sequence has $u_{1} = 8$ and $u_{4} = 27$. The common ratio is positive. Find the common ratio and the sixth term.
Mark Scheme
$u_{4} = 8r^{3} = 27$, so $r^{3} = dfrac{27}{8}$ and $r = dfrac{3}{2}$. [3]
$u_{6} = 8left(dfrac{3}{2}right)^{5} = 8 cdot dfrac{243}{32} = dfrac{243}{4}$. [2]
Total: 5 marks
The sum of the first $n$ terms of an arithmetic series is $S_{n} = 2n^{2} + n$. Find the first term and the common difference.
Question HKC78-AASL-SEQ-03
[Maximum mark: 5]
The sum of the first $n$ terms of an arithmetic series is $S_{n} = 2n^{2} + n$. Find the first term and the common difference.
Mark Scheme
$u_{1} = S_{1} = 2 + 1 = 3$. [1]
$S_{2} = 2(4) + 2 = 10$, so $u_{2} = S_{2} – S_{1} = 7$. [2]
Common difference $d = 7 – 3 = 4$. [1]
In general, $u_{n} = S_{n} – S_{n-1} = 4n – 1$ for $n ge 2$, which has first term $3$ and difference $4$. [1]
Total: 5 marks
