The problem involves finding the total number of levels (tầng) in a pyramid built using coins. Here’s the analysis of the question:

Given:

1. Total coins used: $23,520$ coins.
2. The number of coins in the bottom level (first level): $3,020$.
3. For each higher level, the number of coins decreases by $120$.

Goal:

Determine the total number of levels in the pyramid.

Solution:

The number of coins in each level forms an arithmetic sequence: $a_n = a_1 - (n-1)d$ Where:

• $a_n$: number of coins in the $n$-th level.
• $a_1 = 3020$: number of coins in the first (bottom) level.
• $d = 120$: common difference (the decrease in coins for each level).
• $n$: total number of levels.

The total number of coins used in the pyramid can be expressed as the sum of an arithmetic sequence: $S_n = \frac{n}{2} \left( a_1 + a_n \right)$ Where: $S_n = 23,520$

Substitute $a_n = a_1 - (n-1)d$: $S_n = \frac{n}{2} \left( a_1 + \left( a_1 - (n-1)d \right) \right)$ Simplify: $S_n = \frac{n}{2} \left( 2a_1 - (n-1)d \right)$ $23,520 = \frac{n}{2} \left( 2(3020) - (n-1)(120) \right)$ $23,520 = \frac{n}{2} \left( 6040 - 120n + 120 \right)$ $23,520 = \frac{n}{2} \left( 6160 - 120n \right)$ $23,520 = 3080n - 60n^2$ Rearrange into standard quadratic form: $60n^2 - 3080n + 23,520 = 0$ Simplify: $n^2 - 51.33n + 392 = 0$

Solve the quadratic equation:

Using the quadratic formula: $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here:

• $a = 1$,
• $b = -51.33$,
• $c = 392$.

Substitute: $n = \frac{-(-51.33) \pm \sqrt{(-51.33)^2 - 4(1)(392)}}{2(1)}$ $n = \frac{51.33 \pm \sqrt{2635.47 - 1568}}{2}$ $n = \frac{51.33 \pm \sqrt{1067.47}}{2}$ $n = \frac{51.33 \pm 32.67}{2}$

Choose the positive root: $n = \frac{51.33 + 32.67}{2} = \frac{84}{2} = 42$

Final Answer:

The pyramid has 42 levels.

Let me know if you want further details or clarification! Here are some related questions:

1. How is the formula for the sum of an arithmetic sequence derived?
2. What happens if the total number of coins is slightly different?
3. How do changes in the common difference $d$ affect the solution?
4. What are some real-world applications of arithmetic sequences?
5. Can we solve this problem without using the quadratic equation?

Tip: Always double-check calculations in word problems by substituting the result back into the original equation.