2023 H2 Mathematics Paper 1 Question 2

Equations and Inequalities

Answers

(a)

un=4n3+23n2−46n+29.u_n = 4 n^3 + 23 n^2 - 46 n + 29.

(b)

n≥17.n \geq 17.

Full solutions

(a)

Since unu_n is a cubic polynomial in n,n,

un=an3+bn2+cn+d{u_n = an^3 + bn^2 + cn + d}
u1:a+b+c+d=10u2:8a+4b+2c+d=61u3:27a+9b+3c+d=206u4:64a+16b+4c+d=469\begin{alignat}{2} & u_1: &\quad & a + b + c + d = 10 \\ & u_2: &\quad & 8a + 4b + 2c + d = 61 \\ & u_3: &\quad & 27a + 9b + 3c + d = 206 \\ & u_4: &\quad & 64a + 16b + 4c + d = 469 \end{alignat}

Solving with a GC,

a=4b=23c=−46d=29\begin{align*} a &= 4 \\ b &= 23 \\ c &= - 46 \\ d &= 29 \end{align*}
un=4n3+23n2−46n+29  ■{u_n = 4 n^3 + 23 n^2 - 46 n + 29 \; \blacksquare}

(b)

4n3+23n2−46n+29≥25000{4 n^3 + 23 n^2 - 46 n + 29 \geq 25000}

Using the table in the GC,

u16=21565<25000u17=25546≥25000\begin{align*} u_{16} &= 21565 < 25000 \\ u_{17} &= 25546 \geq 25000 \end{align*}

Hence the range of values of nn is

n≥17,  n∈Z  ■{ n \geq 17, \; n \in \mathbb{Z} \; \blacksquare}

Question Commentary

This is a system of linear equations wrapped in the language of sequences. Algebraic system of linear equations questions have been pretty popular in recent years, including in 2021 P1 Q1, while this technique phrased in sequences was last seen in 2009 P1 Q1.