Series and Sequences

Subject: Additional Mathematics
Topic: 9
Cambridge Code: 4037 / 0606

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Sequences

Sequence - Ordered list of numbers following a pattern

Notation

$a_1, a_2, a_3, \ldots, a_n$

where:

• $a_n$ is the $n$-th term
• $a_1$ is the first term

General Term (Explicit Formula)

Formula for the $n$-th term: $a_n$ or $T_n$

Example: For sequence 3, 6, 9, 12, ... $a_n = 3n$

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Arithmetic Sequences

Arithmetic Sequence - Sequence with constant difference between consecutive terms

Common Difference

$d = a_{n+1} - a_n$

General Formula

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

where:

• $a_n$ is the $n$-th term
• $a_1$ is the first term
• $d$ is the common difference

Example

Sequence: 2, 5, 8, 11, ...

• First term: $a_1 = 2$
• Common difference: $d = 3$
• General formula: $a_n = 2 + (n-1)(3) = 3n - 1$
• 10th term: $a_{10} = 3(10) - 1 = 29$

Sum of Arithmetic Series

$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(2a_1 + (n-1)d)$

where $S_n$ is the sum of first $n$ terms

Example

Sum of first 10 terms of 2, 5, 8, 11, ...:

$S_{10} = \frac{10}{2}(2 + 29) = 5 \times 31 = 155$

Or: $S_{10} = \frac{10}{2}(2(2) + 9(3)) = 5(4 + 27) = 155$

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Geometric Sequences

Geometric Sequence - Sequence with constant ratio between consecutive terms

Common Ratio

$r = \frac{a_{n+1}}{a_n}$

General Formula

$a_n = a_1 \cdot r^{n-1}$

where:

• $a_n$ is the $n$-th term
• $a_1$ is the first term
• $r$ is the common ratio

Example

Sequence: 3, 6, 12, 24, ...

• First term: $a_1 = 3$
• Common ratio: $r = 2$
• General formula: $a_n = 3 \cdot 2^{n-1}$
• 6th term: $a_6 = 3 \cdot 2^5 = 3 \cdot 32 = 96$

Sum of Geometric Series

If $r \neq 1$: $S_n = a_1 \cdot \frac{r^n - 1}{r - 1} = a_1 \cdot \frac{1 - r^n}{1 - r}$

If $r = 1$: $S_n = n \cdot a_1$

Example

Sum of first 6 terms of 3, 6, 12, 24, ...:

$S_6 = 3 \cdot \frac{2^6 - 1}{2 - 1} = 3 \cdot \frac{64 - 1}{1} = 3 \cdot 63 = 189$

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Infinite Geometric Series

Convergent Series - Series with a finite sum (when $|r| < 1$)

Sum to Infinity

$S_\infty = \frac{a_1}{1 - r}, \quad |r| < 1$

Example

Sum of infinite series $2 + 1 + 0.5 + 0.25 + \ldots$

• First term: $a_1 = 2$
• Common ratio: $r = 0.5$

$S_\infty = \frac{2}{1 - 0.5} = \frac{2}{0.5} = 4$

Divergent Series: If $|r| \geq 1$, series diverges (no finite sum)

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Sigma Notation

Sigma Notation - Compact way to write a sum

$\sum_{n=1}^{N} a_n = a_1 + a_2 + a_3 + \cdots + a_N$

Examples

$\sum_{n=1}^{5} 2n = 2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 2+4+6+8+10 = 30$

$\sum_{n=1}^{4} 3^n = 3^1 + 3^2 + 3^3 + 3^4 = 3+9+27+81 = 120$

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Sum Formulas

Sum of Natural Numbers

$\sum_{n=1}^{N} n = \frac{N(N+1)}{2}$

Sum of Squares

$\sum_{n=1}^{N} n^2 = \frac{N(N+1)(2N+1)}{6}$

Sum of Cubes

$\sum_{n=1}^{N} n^3 = \left[\frac{N(N+1)}{2}\right]^2$

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Fibonacci and Other Sequences

Fibonacci Sequence: $F_n = F_{n-1} + F_{n-2}$

Starting with $F_1 = 1, F_2 = 1$: 1, 1, 2, 3, 5, 8, 13, 21, ...

Recursive Sequences

Defined by a recurrence relation: $a_n = f(a_{n-1}, a_{n-2}, \ldots)$

Example: $a_n = 2a_{n-1} + 1$ with $a_1 = 1$

• $a_1 = 1$
• $a_2 = 2(1) + 1 = 3$
• $a_3 = 2(3) + 1 = 7$
• $a_4 = 2(7) + 1 = 15$

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Key Points to Remember

1. Arithmetic: Constant difference $d$
2. Geometric: Constant ratio $r$
3. Arithmetic sum: $S_n = \frac{n}{2}(a_1 + a_n)$
4. Geometric sum: $S_n = a_1 \cdot \frac{1-r^n}{1-r}$
5. Infinite geometric sum: $S_\infty = \frac{a_1}{1-r}$ (if $|r| < 1$)
6. Use sigma notation for compact representation

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Worked Examples

Example 1: Arithmetic Sequence

An arithmetic sequence has $a_3 = 7$ and $a_7 = 15$. Find $a_1$ and $d$:

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

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

Subtract (1) from (2): $8 = 4d \Rightarrow d = 2$

From (1): $7 = a_1 + 4 \Rightarrow a_1 = 3$

Example 2: Geometric Sum

A geometric series has $a_1 = 2$ and $r = \frac{1}{2}$. Find $S_5$:

$S_5 = 2 \cdot \frac{1 - (\frac{1}{2})^5}{1 - \frac{1}{2}} = 2 \cdot \frac{1 - \frac{1}{32}}{\frac{1}{2}} = 2 \cdot 2 \cdot \frac{31}{32} = \frac{31}{8} = 3.875$

Example 3: Infinite Series

Find sum of infinite geometric series: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$

• $a_1 = \frac{1}{2}$, $r = \frac{1}{2}$

$S_\infty = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1$

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Practice Questions

1. Find the general term and 20th term of: 5, 10, 15, 20, ...

2. Find the sum of first 8 terms of: 3, 9, 27, 81, ...

3. Find the sum to infinity of: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$

4. Given arithmetic sequence with $a_2 = 5$ and $a_5 = 14$, find $a_{10}$.

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Revision Tips

• Arithmetic: common difference $d$ (add/subtract)
• Geometric: common ratio $r$ (multiply/divide)
• Two equations needed to find $a_1$ and $d$
• Check if geometric series converges before finding $S_\infty$
• Use sigma notation for multiple terms