Sequence and Series: Types, Sums, and Applications

TAKEAWAYS:

• A sequence is an ordered list of numbers (e.g., 2, 4, 6, ...), while a series is the sum of its terms (e.g., 2 + 4 + 6 + ...).
• Arithmetic sequences grow by a constant difference (e.g., 3, 7, 11, ...; d = 4), and their sum is $S_n=\frac{n}{2}[2a+(n-1)d]$.
• Geometric sequences multiply by a constant ratio (e.g., 5, 15, 45, ...; r = 3), and their sum is $S_n=a\frac{1-r^n}{1-r}$ (if r ≠ 1).
• Special series like infinite geometric series converge only if |r| < 1, with sum $S=\frac{a}{1-r}$.
• Applications include calculating loan payments, population growth, and financial investments.

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## What is a Sequence?
A **sequence** is a list of numbers written in a specific order. Each number in the sequence is called a **term**. Sequences can be finite (limited number of terms) or infinite (unlimited terms).

### Types of Sequences
1. **Finite Sequence**: Has a fixed number of terms (e.g., 1, 3, 5, 7, 9).
2. **Infinite Sequence**: Goes on forever (e.g., 2, 4, 6, 8, ...).

### General Form of a Sequence
The *n*-th term of a sequence is often written as \( a_n \). For example:
- If \( a_n = 2n \), the sequence is 2, 4, 6, 8, ... (each term is twice its position).

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## Arithmetic Sequence
An **arithmetic sequence** is a sequence where each term increases or decreases by a constant difference (*d*).

### Formula for the *n*-th Term
The *n*-th term of an arithmetic sequence is given by:
\[
a_n = a + (n - 1)d
\]
where:
- \( a \) = first term,
- \( d \) = common difference,
- \( n \) = term number.

### Example 1: Find the 10th term of 3, 7, 11, 15, ...
Here, \( a = 3 \) and \( d = 4 \).
\[
a_{10} = 3 + (10 - 1) \times 4 = 3 + 36 = 39
\]
**Answer**: The 10th term is **39**.

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## Sum of an Arithmetic Sequence
The sum of the first *n* terms (\( S_n \)) of an arithmetic sequence is:
\[
S_n = \frac{n}{2} [2a + (n - 1)d]
\]
or equivalently:
\[
S_n = \frac{n}{2} (a + l)
\]
where \( l \) = last term.

### Example 2: Sum of the first 10 terms of 5, 12, 19, ...
Here, \( a = 5 \), \( d = 7 \), and \( n = 10 \).
First, find the 10th term:
\[
a_{10} = 5 + (10 - 1) \times 7 = 5 + 63 = 68
\]
Now, calculate the sum:
\[
S_{10} = \frac{10}{2} (5 + 68) = 5 \times 73 = 365
\]
**Answer**: The sum is **365**.

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## Geometric Sequence
A **geometric sequence** is a sequence where each term is multiplied by a constant ratio (*r*).

### Formula for the *n*-th Term
The *n*-th term of a geometric sequence is:
\[
a_n = a \times r^{n-1}
\]
where:
- \( a \) = first term,
- \( r \) = common ratio,
- \( n \) = term number.

### Example 3: Find the 6th term of 2, 6, 18, 54, ...
Here, \( a = 2 \) and \( r = 3 \).
\[
a_6 = 2 \times 3^{6-1} = 2 \times 243 = 486
\]
**Answer**: The 6th term is **486**.

---

## Sum of a Geometric Sequence
The sum of the first *n* terms of a geometric sequence is:
\[
S_n = a \frac{1 - r^n}{1 - r} \quad \text{(if } r \neq 1\text{)}
\]

### Example 4: Sum of the first 5 terms of 3, 12, 48, ...
Here, \( a = 3 \), \( r = 4 \), and \( n = 5 \).
\[
S_5 = 3 \frac{1 - 4^5}{1 - 4} = 3 \frac{1 - 1024}{-3} = 3 \times 341 = 1023
\]
**Answer**: The sum is **1023**.

---

## Infinite Geometric Series
An **infinite geometric series** converges (has a finite sum) only if \( |r| < 1 \). Its sum is:
\[
S = \frac{a}{1 - r}
\]

### Example 5: Sum of \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \)
Here, \( a = 1 \) and \( r = \frac{1}{2} \).
\[
S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2
\]
**Answer**: The sum is **2**.

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## Applications of Sequences and Series
1. **Finance**: Calculating loan repayments or compound interest.
2. **Physics**: Modeling wave patterns or radioactive decay.
3. **Biology**: Population growth or drug dosage schedules.

### Example 6: Compound Interest
If you invest ₹10,000 at 5% interest compounded annually, how much will you have after 3 years?
Here, \( a = 10000 \), \( r = 1.05 \), and \( n = 3 \).
\[
A = 10000 \times (1.05)^3 = 10000 \times 1.157625 = 11576.25
\]
**Answer**: You will have **₹11,576.25** after 3 years.

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## Comparison Table: Arithmetic vs. Geometric Sequences

| Feature               | Arithmetic Sequence               | Geometric Sequence               |
|-----------------------|-----------------------------------|----------------------------------|
| **Definition**        | Difference between terms is constant | Ratio between terms is constant |
| **Formula for *n*-th term** | \( a_n = a + (n-1)d \)       | \( a_n = a \times r^{n-1} \)     |
| **Sum of *n* terms**  | \( S_n = \frac{n}{2} [2a + (n-1)d] \) | \( S_n = a \frac{1 - r^n}{1 - r} \) |
| **Example**           | 2, 5, 8, 11, ... (*d* = 3)       | 3, 6, 12, 24, ... (*r* = 2)       |
| **Infinite Sum**      | Diverges (no finite sum)          | Converges if \( |r| < 1 \)       |

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## NEB Board-Style Questions

### Short Answer Questions
1. Write the first 4 terms of the sequence defined by \( a_n = 2n + 1 \).
2. Find the 8th term of the arithmetic sequence: 7, 13, 19, 25, ...
3. Calculate the sum of the first 6 terms of the geometric sequence: 5, 15, 45, ...

### Long Answer Questions
4. The sum of the first 10 terms of an arithmetic sequence is 155, and the first term is 3. Find the common difference.
5. A geometric sequence has a first term of 4 and a common ratio of 2. Find the sum of the first 6 terms.
6. A ball is dropped from a height of 10 meters. It bounces back to 60% of its previous height each time. How far has the ball traveled after 5 bounces?

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## Exam Tip
- **Memorize formulas**: Know the *n*-th term and sum formulas for both arithmetic and geometric sequences.
- **Practice calculations**: Always double-check your arithmetic when calculating sums or terms.
- **Identify the type**: Before solving, determine whether the sequence is arithmetic or geometric.
- **Watch for infinite series**: Only geometric series with \( |r| < 1 \) have finite sums.
- **Real-world applications**: Questions often involve money (interest, loans) or physics (motion, waves). Relate the problem to sequences/series.

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```mermaid
flowchart TD
    A["Sequence"] --> B["Arithmetic"]
    A --> C["Geometric"]
    B --> D["Common Difference (d)"]
    C --> E["Common Ratio (r)"]
    D --> F["Sum: $S_n=\frac{n}{2}[2a+(n-1)d]$"]
    E --> G["Sum: $S_n=a\frac{1-r^n}{1-r}$"]
    E --> H["Infinite Sum: $S=\frac{a}{1-r}$ if $|r|<1$"]