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Arithmetic Sequence: Formula & Summation

CSCA Arithmetic Sequence: Formula & Summation study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Arithmetic Sequence: Formula & Summation

1. Core Concepts

**Definition**: A sequence is called an **Arithmetic Sequence** if the difference between any term (from the 2nd term) and its preceding term is a constant. This constant is called the **Common Difference ($d$)**.

**Mathematical Expression**:

$$a_{n} - a_{n-1} = d \quad (n \ge 2)$$

**Functional Perspective**:

The general term $a_n = dn + (a_1 - d)$ is a linear function of $n$. The graph of an arithmetic sequence consists of **discrete points** lying on the line $y = dx + (a_1-d)$. The common difference $d$ is the **slope**.

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2. Key Formulas

#### (1) General Term Formula

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

* **Generalized Form**: $a_n = a_m + (n-m)d$ (Useful for finding terms without needing $a_1$ first).

#### (2) Summation Formulas ($S_n$)

Let $S_n = a_1 + a_2 + \dots + a_n$.

* **Formula 1 (First & Last Term)**:

$$S_n = \frac{n(a_1 + a_n)}{2}$$

*Intuition*: (First + Last) $\times$ Count $\div 2$. Similar to the area of a trapezoid.

* **Formula 2 (First Term & Difference)**:

$$S_n = na_1 + \frac{n(n-1)}{2}d$$

**Derivation Logic (Reverse and Add)**:

Write the sum forwards, then write it backwards underneath. Add them vertically. Each column sums to $(a_1 + a_n)$. This method is often tested in CSCA.

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3. Key Properties

Using properties is often faster than formulas:

1. **Arithmetic Mean**: If $a, b, c$ form an arithmetic sequence, then $2b = a + c$.

2. **Index Sum Property**: If $m + n = p + q$, then $a_m + a_n = a_p + a_q$.

* E.g., $a_1 + a_{10} = a_3 + a_8$.

* Special case: If $m+n=2k$, then $a_m + a_n = 2a_k$.

3. **Partial Sum Property**: $S_n, S_{2n}-S_n, S_{3n}-S_{2n}, \dots$ also form an arithmetic sequence (Difference is $n^2d$).

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4. Typical Examples

**Example 1: Using Generalized Formula**

Given $a_3 = 8$ and $a_7 = 20$, find $a_{15}$.

**Solution**:

Use $a_7 = a_3 + (7-3)d$.

$20 = 8 + 4d \Rightarrow 4d = 12 \Rightarrow d = 3$.

Then $a_{15} = a_7 + (15-7)d = 20 + 8 \times 3 = 44$.

**Example 2: Using Index Sum Property**

Given $a_4 + a_8 = 16$ in an arithmetic sequence, find $S_{11}$.

**Solution**:

Formula: $S_{11} = \frac{11(a_1 + a_{11})}{2}$.

Since $1 + 11 = 4 + 8 = 12$, we have $a_1 + a_{11} = a_4 + a_8 = 16$.

Substitute: $S_{11} = \frac{11 \times 16}{2} = 88$.

**Example 3: Reverse and Add Application**

Let $f(x) = \frac{1}{1+2^x}$. Find sum $S = f(-5) + \dots + f(5)$.

**Solution**:

Notice $f(x) + f(-x) = 1$. This pairs terms like an arithmetic sum.

Pairs: $(-5,5), (-4,4) \dots$. There are 5 pairs and $f(0)$.

Sum $= 5 \times 1 + f(0) = 5 + 0.5 = 5.5$.

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5. Common Pitfalls

* **Counting Terms**: When calculating $S_n$, ensure $n$ is correct. From 3rd to 8th term is $8-3+1=6$ terms.

* **Confusion**: Don't confuse the $n$-th term ($a_n$) with the sum of $n$ terms ($S_n$).

* **Negative Difference**: Remember $d$ can be negative (decreasing sequence).