等差数列děngchā shùliè

Core Concept

An arithmetic sequence (等差数列) is one of the most fundamental types of sequences in mathematics. Starting from the second term, the difference between any term and its preceding term equals the same constant, called the common difference (公差), typically denoted by $d$.

Mathematical Definition

For a sequence $\{a_n\}$, if there exists a constant $d$ such that:

$a_{n+1} - a_n = d \quad (n \in \mathbb{N}^*)$

Then the sequence $\{a_n\}$ is called an arithmetic sequence, with $d$ as the common difference.

General Term Formula

The $n$-th term of an arithmetic sequence can be expressed using the first term $a_1$ and common difference $d$:

$a_n = a_1 + (n-1)d$

Derivation:

• $a_2 = a_1 + d$
• $a_3 = a_2 + d = a_1 + 2d$
• $a_4 = a_3 + d = a_1 + 3d$
• ...
• $a_n = a_1 + (n-1)d$

Sum Formula

The sum of the first $n$ terms has two common formulas:

Formula 1 (using first and last terms): $S_n = \frac{n(a_1 + a_n)}{2}$

Formula 2 (using first term and common difference): $S_n = na_1 + \frac{n(n-1)}{2}d = \frac{n[2a_1 + (n-1)d]}{2}$

Important Properties

Property 1: Arithmetic Mean

If $a$, $b$, $c$ form an arithmetic sequence, then: $b = \frac{a + c}{2}$

That is, $b$ is the arithmetic mean of $a$ and $c$.

Property 2: Index Property

If $m + n = p + q$ (where $m, n, p, q \in \mathbb{N}^*$), then: $a_m + a_n = a_p + a_q$

In particular, if $m + n = 2p$, then: $a_m + a_n = 2a_p$

Property 3: Sum Property

The sum of first $n$ terms can be viewed as $n$ times the average of the first and last terms: $S_n = \frac{n(a_1 + a_n)}{2}$

Real-World Applications

Application 1: Bank Savings

Problem: Ming saves 500 yuan every month for 12 months. What is the total savings after 12 months?

Analysis: Monthly deposits form an arithmetic sequence with $a_1 = 500$, $d = 500$, $n = 12$.

Solution: $S_{12} = \frac{12 \times (500 + 6500)}{2} = 42,000 \text{ yuan}$

Application 2: Theater Seating

Problem: A theater has 20 seats in the first row. Each subsequent row has 2 more seats than the previous row. If there are 30 rows, how many seats are there in total?

Analysis:

• First term $a_1 = 20$
• Common difference $d = 2$
• Number of terms $n = 30$

Solution: $S_{30} = 30 \times 20 + \frac{30 \times 29}{2} \times 2 = 600 + 870 = 1,470 \text{ seats}$

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus and Chinese standardized test formats to help students familiarize themselves with question types and problem-solving approaches.

Example 1: Basic (Difficulty ★★☆☆☆)

In an arithmetic sequence $\{a_n\}$, $a_3 = 7$ and $a_7 = 15$. Find $a_{10}$.

Options:

• A. 19
• B. 21
• C. 23
• D. 25

Detailed Solution:

Method 1: Using the general term formula

1. From the general term formula:

   • $a_3 = a_1 + 2d = 7$ ... ①
   • $a_7 = a_1 + 6d = 15$ ... ②

2. ② - ① gives: $4d = 8$, so $d = 2$

3. Substituting into ①: $a_1 + 4 = 7$, so $a_1 = 3$

4. $a_{10} = a_1 + 9d = 3 + 18 = 21$

Answer: B

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Example 2: Intermediate (Difficulty ★★★☆☆)

In an arithmetic sequence $\{a_n\}$, $S_5 = 25$ and $S_{10} = 100$. Find $S_{15}$.

Options:

• A. 175
• B. 200
• C. 225
• D. 250

Detailed Solution:

Method: Using the property that $S_5$, $S_{10} - S_5$, and $S_{15} - S_{10}$ also form an arithmetic sequence:

• $S_5 = 25$
• $S_{10} - S_5 = 100 - 25 = 75$
• Common difference: $75 - 25 = 50$

Therefore: $S_{15} - S_{10} = 75 + 50 = 125$

$S_{15} = 100 + 125 = 225$

Answer: C

Common Mistakes

❌ Mistake 1: Can the common difference be zero?

Answer: Mathematically, when $d = 0$, the sequence is a constant sequence, which is technically an arithmetic sequence. However, in CSCA exams, "arithmetic sequence" usually implies $d \neq 0$ unless specified otherwise.

❌ Mistake 2: Is an arithmetic sequence always increasing?

Answer: Not necessarily!

• $d > 0$ → increasing sequence
• $d < 0$ → decreasing sequence
• $d = 0$ → constant sequence

❌ Mistake 3: Confusing with geometric sequences

Key Difference:

• Arithmetic sequence: difference between consecutive terms is constant ($a_{n+1} - a_n = d$)
• Geometric sequence: ratio between consecutive terms is constant ($\frac{a_{n+1}}{a_n} = r$)

Study Tips

1. ✅ Master the formulas - General term and sum formulas are fundamental
2. ✅ Understand common difference - It can be positive, negative, or zero
3. ✅ Practice properties - Especially arithmetic mean and index properties
4. ✅ Solve various problems - Arithmetic sequences often combine with functions and inequalities
5. ✅ Compare with geometric sequences - Understand the differences clearly

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💡 Exam Tip: Arithmetic sequences are a high-frequency topic in CSCA math exams, accounting for about 60% of sequence questions. Practice 2-3 related problems daily to ensure mastery.