瑞安有一些纸牌，他把它们叠成金字塔。他首先叠起纸牌组成一个 1 层的金字塔，然后是另一个 2 层的金字塔，依此类推。不同层数 n 的金字塔如下图所示： 设 $S_n$ 表示一个 $n$ 层金字塔中的纸牌数量。已知对于某些常数 $a,b$ 和 $c$，有 $S_n = an^2 + bn + c$。

1. Identify that the number of additional cards needed is the difference between the total cards in the $n$th pyramid and the $(n-1)$th pyramid.

$\text{Additional cards} = S_n - S_{n-1}$

2. Substitute the algebraic expressions for $S_n$ and $S_{n-1}$ and simplify.

1. Determine the number of cards $S_1$, $S_2$, and $S_3$ by logically counting the cards from the described pyramids.

By counting the cards from the structural pattern of the pyramids:
Level 1 ($n=1$): 2 leaning cards.
$S_1 = 2$

Level 2 ($n=2$): The previous pyramid, plus 1 flat base card, plus 4 leaning cards at the bottom.
$S_2 = 2 + 1 + 4 = 7$

Level 3 ($n=3$): The previous pyramid, plus 2 flat base cards, plus 6 leaning cards at the bottom.
$S_3 = 7 + 2 + 6 = 15$

2. Formulate a system of linear equations by substituting $n=1, 2, 3$ into the given quadratic expression.

Substitute $n=1, 2, 3$ into the given equation $S_n = an^2 + bn + c$:

For $n=1$:

For $n=2$:

For $n=3$:

3. Solve the system of equations to find the values of $a$, $b$, and $c$.

Subtract (1) from (2) to eliminate $c$:
$(4a + 2b + c) - (a + b + c) = 7 - 2$
$3a + b = 5 \quad \text{--- (4)}$

Subtract (2) from (3) to eliminate $c$:
$(9a + 3b + c) - (4a + 2b + c) = 15 - 7$
$5a + b = 8 \quad \text{--- (5)}$

Subtract (4) from (5) to eliminate $b$:
$(5a + b) - (3a + b) = 8 - 5$
$2a = 3 \implies a = 1.5$

Substitute $a = 1.5$ into (4):
$3(1.5) + b = 5$
$4.5 + b = 5 \implies b = 0.5$

Substitute $a = 1.5$ and $b = 0.5$ into (1):
$1.5 + 0.5 + c = 2$
$2 + c = 2 \implies c = 0$

1. Identify the general term (cards in the nth level) from part (a)

From part (a), the number of additional cards needed to go from the $(n-1)$th to the $n$th level is the number of cards in the $n$th level:
$u_n = S_n - S_{n-1} = 2an - a + b.$
Thus $S_n = u_1 + u_2 + \cdots + u_n$, i.e. $S_n$ is a sum of the terms $u_n$.

2. Show consecutive terms have a constant difference and state it

Consider the difference between consecutive terms:
$u_n - u_{n-1} = (2an - a + b) - \big(2a(n-1) - a + b\big) = 2a.$
Since this difference is constant for all $n$, the terms $u_n$ form an arithmetic progression, and therefore $S_n = u_1 + u_2 + \cdots + u_n$ is the sum of an arithmetic progression.

Using $a = \tfrac{3}{2}$ (from part (ii)), the common difference is
$2a = 2\left(\tfrac{3}{2}\right) = 3.$

1. Set up the summation for the total number of cards across the 23 pyramids

Using the sequence expression $S_n = 1.5n^2 + 0.5n$ established in the earlier parts, the total number of cards required to build one pyramid of each level from 1 to 23 is the sum of $S_n$:
$\text{Total cards} = \sum_{n=1}^{23} S_n = \sum_{n=1}^{23} (1.5n^2 + 0.5n)$

2. Evaluate the summation

Evaluating this finite sum (using a Graphing Calculator or standard formulas for $\sum n^2$ and $\sum n$):
$\text{Total cards} = 6624$