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Introduction to Sequences

A sequence is an ordered list of numbers that follow a specific pattern. Each number in a sequence is called a term. We typically denote the terms of a sequence using subscript notation:

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

A sequence can be defined by an explicit formula that gives the value of the

n

-th term directly, or by a recursive formula that defines each term based on previous terms.

Types of Sequences

Explicit Formula

a_n = f(n)

Where

f(n)

is a function that directly computes the

n

-th term.

Recursive Formula

a_1 = \text{initial value}

a_n = g(a_1, a_2, \ldots, a_{n-1})

Where

g

defines each term based on previous terms.

A series is the sum of the terms of a sequence. If

\{a_n\}

is a sequence, then the corresponding series is written as:

\sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + \cdots + a_n

Arithmetic Sequences

An arithmetic sequence is a sequence where each term differs from the previous term by a constant value called the common difference (

d

).

a_n = a_1 + (n-1)d

In an arithmetic sequence:

•

a_1

is the first term

•

d

is the common difference:

d = a_2 - a_1 = a_3 - a_2 = \ldots = a_{n+1} - a_n

• The sequence increases if

d > 0

and decreases if

d < 0

Problem: Find the 20th term of the arithmetic sequence: 5, 8, 11, 14, ...

1

First, identify the first term

a_1

and the common difference

d

:

a_1 = 5

d = a_2 - a_1 = 8 - 5 = 3

2

Use the formula

a_n = a_1 + (n-1)d

to find the 20th term:

a_{20} = 5 + (20-1)3 = 5 + 19 \cdot 3 = 5 + 57 = 62

3

Therefore, the 20th term of the sequence is 62.

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence. For a finite arithmetic sequence with

n

terms, the sum

S_n

can be calculated using:

Sum of an Arithmetic Series

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

S_n = \frac{n}{2}[2a_1 + (n-1)d]

Where

a_1

is the first term,

a_n

is the nth term,

n

is the number of terms, and

d

is the common difference.

Derivation of the Sum Formula

1

Write out the sum in two different ways:

S_n = a_1 + (a_1 + d) + (a_1 + 2d) + \ldots + (a_1 + (n-1)d)

S_n = a_n + (a_n - d) + (a_n - 2d) + \ldots + (a_n - (n-1)d)

2

Add these two equations:

2S_n = [a_1 + a_n] + [(a_1 + d) + (a_n - d)] + \ldots + [(a_1 + (n-1)d) + (a_n - (n-1)d)]

3

Simplify each pair of terms:

2S_n = [a_1 + a_n] + [a_1 + a_n] + \ldots + [a_1 + a_n] = n(a_1 + a_n)

4

Solve for

S_n

:

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

5

Since

a_n = a_1 + (n-1)d

, we can rewrite the formula:

S_n = \frac{n}{2}(a_1 + a_1 + (n-1)d) = \frac{n}{2}[2a_1 + (n-1)d]

Problem: Find the sum of the first 30 positive integers.

1

This is an arithmetic sequence with

a_1 = 1

,

d = 1

, and

n = 30

.

2

The 30th term is

a_{30} = 1 + (30-1)1 = 30

3

Using the formula

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

:

S_{30} = \frac{30}{2}(1 + 30) = 15 \cdot 31 = 465

4

Therefore, the sum of the first 30 positive integers is 465.

Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed non-zero number called the common ratio (

r

).

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

In a geometric sequence:

•

a_1

is the first term

•

r

is the common ratio:

r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \ldots = \frac{a_{n+1}}{a_n}

• The sequence increases in magnitude if

|r| > 1

and decreases if

|r| < 1

• If

r > 0

, all terms have the same sign as

a_1

• If

r < 0

, the terms alternate in sign

Problem: Find the 8th term of the geometric sequence: 3, 6, 12, 24, ...

1

First, identify the first term

a_1

and the common ratio

r

:

a_1 = 3

r = \frac{a_2}{a_1} = \frac{6}{3} = 2

2

Use the formula

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

to find the 8th term:

a_8 = 3 \cdot 2^{8-1} = 3 \cdot 2^7 = 3 \cdot 128 = 384

3

Therefore, the 8th term of the sequence is 384.

Geometric Series

A geometric series is the sum of the terms in a geometric sequence. For a finite geometric sequence with

n

terms and common ratio

r \neq 1

, the sum

S_n

can be calculated using:

Sum of a Geometric Series

S_n = \frac{a_1(1-r^n)}{1-r}

S_n = \frac{a_1(r^n-1)}{r-1}

Where

a_1

is the first term,

r

is the common ratio, and

n

is the number of terms.
The first formula is commonly used when

|r| < 1

and the second when

|r| > 1

.

Derivation of the Sum Formula

1

Write out the sum:

S_n = a_1 + a_1r + a_1r^2 + \ldots + a_1r^{n-1}

2

Multiply both sides by

r

:

rS_n = a_1r + a_1r^2 + a_1r^3 + \ldots + a_1r^n

3

Subtract the second equation from the first:

S_n - rS_n = a_1 - a_1r^n

4

Factor out

S_n

on the left side:

S_n(1-r) = a_1(1-r^n)

5

Solve for

S_n

:

S_n = \frac{a_1(1-r^n)}{1-r}

6

For

|r| > 1

, we can rewrite this as:

S_n = \frac{a_1(r^n-1)}{r-1}

Problem: Find the sum of the first 10 terms of the geometric series: 5, 15, 45, 135, ...

1

First, identify the first term

a_1

and the common ratio

r

:

a_1 = 5

r = \frac{a_2}{a_1} = \frac{15}{5} = 3

2

Since

r = 3 > 1

, use the formula

S_n = \frac{a_1(r^n-1)}{r-1}

to find the sum:

S_{10} = \frac{5(3^{10}-1)}{3-1} = \frac{5(59,049-1)}{2} = \frac{5 \cdot 59,048}{2} = 147,620

3

Therefore, the sum of the first 10 terms is 147,620.

Infinite Geometric Series

When

|r| < 1

, the sum of an infinite geometric series converges to a finite value:

\sum_{i=0}^{\infty} ar^i = \frac{a}{1-r} \quad \text{for } |r| < 1

This formula has many applications in calculus and real-world scenarios, such as calculating repeating decimals or determining the total distance traveled by a bouncing ball.

Problem: Express the repeating decimal

0.999...

as a fraction.

1

Let

S = 0.999...

represent the sum of the infinite geometric series.

2

We can write this as:

S = 0.9 + 0.09 + 0.009 + ...

S = 9 \cdot 0.1 + 9 \cdot 0.01 + 9 \cdot 0.001 + ...

S = 9 \cdot (0.1 + 0.01 + 0.001 + ...)

3

The sum in parentheses is an infinite geometric series with

a = 0.1

and

r = 0.1

:

0.1 + 0.01 + 0.001 + ... = \frac{0.1}{1-0.1} = \frac{0.1}{0.9} = \frac{1}{9}

4

Therefore:

S = 9 \cdot \frac{1}{9} = 1

5

So

0.999... = 1

Applications of Sequences and Series

Sequences and series have numerous applications in mathematics and real-world scenarios.

Compound Interest

If

P

is the principal,

r

is the annual interest rate, and interest is compounded

n

times a year, the amount after

t

years is:

A = P\left(1 + \frac{r}{n}\right)^{nt}

This is a geometric sequence with common ratio

\left(1 + \frac{r}{n}\right)^n

.

Population Growth

A population that grows by a constant percentage follows a geometric sequence:

P_n = P_0(1 + r)^n

Where

P_0

is the initial population,

r

is the growth rate, and

n

is the number of time periods.

Summation of Natural Numbers

The sum of the first

n

natural numbers:

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

This is an arithmetic series with

a_1 = 1

and

d = 1

.

Problem: A ball is dropped from a height of 10 meters. Each time it hits the ground, it rebounds to 60% of its previous height. Find the total distance the ball travels.

1

Let's break this down into up and down movements:

• Initial drop: 10 meters down

• First rebound:

10 \cdot 0.6 = 6

meters up

• Second drop: 6 meters down

• Second rebound:

6 \cdot 0.6 = 3.6

meters up

And so on...

2

The total distance is the sum of all these distances:

D = 10 + 6 + 6 + 3.6 + 3.6 + ...

3

Grouping the up and down movements:

D = 10 + (6 + 6) + (3.6 + 3.6) + ...

D = 10 + 12 + 7.2 + ...

4

After the initial 10 meters, we have pairs of distances that form a geometric sequence:

D = 10 + (10 \cdot 0.6 \cdot 2) + (10 \cdot 0.6^2 \cdot 2) + ...

D = 10 + 10 \cdot 2 \cdot (0.6 + 0.6^2 + 0.6^3 + ...)

5

The sum in parentheses is an infinite geometric series with

a = 0.6

and

r = 0.6

:

0.6 + 0.6^2 + 0.6^3 + ... = \frac{0.6}{1-0.6} = \frac{0.6}{0.4} = 1.5

6

Therefore:

D = 10 + 10 \cdot 2 \cdot 1.5 = 10 + 30 = 40

7

The ball travels a total distance of 40 meters.

Sequences and Series

Exponents and Logarithms

Functions

Polynomial Functions

Rational Functions

Exponential and Log Functions

Trigonometric Functions

Trigonometric Identities

Solving Triangles

Vectors

Limits and Continuity

Derivatives

Integrals

Descriptive Statistics

Probability Concepts

Probability Distributions

Complex Numbers

Further Calculus

Matrices

Applications and Modeling

IA Preparation (Math)

IA Preparation (Physics)

Sequences and Series

Introduction to Sequences

Types of Sequences

Arithmetic Sequences

Arithmetic Series

Sum of an Arithmetic Series

Geometric Sequences

Geometric Series

Sum of a Geometric Series

Infinite Geometric Series

Applications of Sequences and Series