2024 H2 Mathematics Paper 2 Question 4

Arithmetic and Geometric Progressions (APs, GPs)

Answers

(a)

$t_{n} = 6 n^{2} - 22 n + 6.$

(b)

$n = 5$ or $n = 7.$

(c)

$196.$

(d)

(i)

If $n$ is even, then $3 n^{2}$ and $- 5 n$ are even so $w_{n} = 3 n^{2} - 5 n + 7$ is odd.
If $n$ is odd, then $3 n^{2}$ and $- 5 n$ are odd so $w_{n} = 3 n^{2} - 5 n + 7$ is odd.
Hence all the terms in series $W$ are odd.

(ii)

All the terms in $u_{n} = 2 (25 n - 102)$ are even. Hence series $U$ and $W$ do not have any terms in common.

Full Solutions

(a)

\begin{aligned} t_{n} & = S_{n} - S_{n - 1} \\ = 2 n^{3} - 8 n^{2} - 4 n - (2 {(n - 1)}^{3} - 8 {(n - 1)}^{2} - 4 (n - 1)) \\ = 2 n^{3} - 8 n^{2} - 4 n - 2 {(n - 1)}^{3} + 8 {(n - 1)}^{2} + 4 n - 4 \\ = 2 n^{3} - 8 n^{2} - 2 {(n - 1)}^{3} + 8 {(n - 1)}^{2} - 4 \\ = 2 n^{3} - 8 n^{2} - 2 (n^{3} - 3 n^{2} + 3 n - 1) + 8 (n^{2} - 2 n + 1) - 4 \\ = 2 n^{3} - 8 n^{2} - 2 n^{3} + 6 n^{2} - 6 n - 2 (- 1) + 8 n^{2} - 16 n + 8 (1) - 4 \\ = 6 n^{2} - 22 n + 6 \end{aligned}

(b)

\begin{matrix} u_{n} = t_{n} \\ 50 n - 204 = 6 n^{2} - 22 n + 6 \\ 6 n^{2} - 72 n + 210 = 0 \\ n^{2} - 12 n + 35 = 0 \\ (n - 5) (n - 7) = 0 \\ n = 5 ∎ or n = 7 ∎ \end{matrix}

(c)

\begin{aligned} u_{n} & = v_{m} \\ 50 n - 204 & = 3 m + 16 \\ 3 m & = 50 n - 220 \\ m & = \frac{50 n - 220}{3} \end{aligned}

We want both $m$ and $n$ to be positive integers, and $u_{n} > 100$

Using the table in the GC,

$n$	$m = \frac{50 n - 220}{3}$	$u_{n} = 50 n - 204$
5	$10$ ✅	$46$ ❌
6	$\frac{80}{3}$ ❌	$96$ ❌
7	$\frac{130}{3}$ ❌	$146$ ✅
8	60 ✅	196 ✅

The smallest number greater than $100$ that is in both series $U$ and $V$ is

u_{8} = v_{60} = 196 ∎

(d)

(i)

If $n$ is even, then $3 n^{2}$ and $- 5 n$ are even so $w_{n} = 3 n^{2} - 5 n + 7$ is odd

If $n$ is odd, then $3 n^{2}$ and $- 5 n$ are odd so $w_{n} = 3 n^{2} - 5 n + 7$ is odd

Hence all the terms in series $W$ are odd $∎$

(ii)

We observe that all the terms in

\begin{aligned} u_{n} & = 50 n - 204 \\ = 2 (25 n - 102) \end{aligned}

are even. Hence series $U$ and $W$ do not have any terms in common $∎$