NYJC Promo 9758/2020/Q2

1. Define the general form of the cubic polynomial for the sequence.

Let $u_n = an^3 + bn^2 + cn + d$, where $a, b, c$, and $d$ are constants.

2. Substitute the given values to form a system of linear equations.

Using the given terms:
For $n=1$, $a(1)^3 + b(1)^2 + c(1) + d = 32.1 \implies a + b + c + d = 32.1$
For $n=2$, $a(2)^3 + b(2)^2 + c(2) + d = 17 \implies 8a + 4b + 2c + d = 17$
For $n=3$, $a(3)^3 + b(3)^2 + c(3) + d = 0.7 \implies 27a + 9b + 3c + d = 0.7$
For $n=4$, $a(4)^3 + b(4)^2 + c(4) + d = -7.8 \implies 64a + 16b + 4c + d = -7.8$

3. Solve the system using a graphing calculator to find the explicit formula.

Solving the system of linear equations using a graphing calculator gives:
$a = 1.5$
$b = -9.6$
$c = 3.2$
$d = 37$

Therefore, $u_n = 1.5n^3 - 9.6n^2 + 3.2n + 37$.

1. Formulate the sequence $u_n$ from the conditions in the earlier part and set up the required inequality.

$u_n = an^3 + bn^2 + cn + d$
Substituting $n=1, 2, 3, 4$ and their corresponding values $u_1 = 32.1$, $u_2 = 17$, $u_3 = 0.7$, $u_4 = -7.8$ gives a system of linear equations:
$\begin{cases} a+b+c+d = 32.1 \\ 8a+4b+2c+d = 17 \\ 27a+9b+3c+d = 0.7 \\ 64a+16b+4c+d = -7.8 \end{cases}$
Solving this system (e.g., using a graphing calculator) yields $a=1.5$, $b=-9.6$, $c=3.2$, $d=37$.
Hence, the general term is $u_n = 1.5n^3 - 9.6n^2 + 3.2n + 37$.

We require $u_n > 555$:
$1.5n^3 - 9.6n^2 + 3.2n + 37 > 555$
$1.5n^3 - 9.6n^2 + 3.2n - 518 > 0$

2. Solve the cubic inequality to find the least integer $n$.

Using a graphing calculator to solve $1.5n^3 - 9.6n^2 + 3.2n - 518 = 0$, we find the real root is $n \approx 9.79$.
Since the leading coefficient is positive, the inequality is satisfied for $n > 9.79$.
Since $n$ must be a positive integer, the least value is $10$.

We can optionally verify this by checking integer values:
$u_9 = 1.5(9)^3 - 9.6(9)^2 + 3.2(9) + 37 = 381.7 \le 555$
$u_{10} = 1.5(10)^3 - 9.6(10)^2 + 3.2(10) + 37 = 609 > 555$
Therefore, the least value of $n$ is 10.