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Introduction to Sequences

A sequence is an ordered list of numbers following a specific pattern. Understanding sequences and their sums (series) is fundamental to calculus, statistics, and many real-world applications like finance and computer science.

Sequence

A sequence is a function whose domain is the set of positive integers. We write sequences as:

a₁, a₂, a₃, ..., aₙ, ... or {aₙ}

Each aₙ is called a term, and n is the index or position.

Arithmetic Sequences

Arithmetic Sequence

An arithmetic sequence has a constant difference between consecutive terms.

aₙ = a₁ + (n - 1)d

a₁ = first term
d = common difference (aₙ₊₁ - aₙ)
n = position/term number

Example: 3, 7, 11, 15, 19, ... (a₁ = 3, d = 4)

Geometric Sequences

Geometric Sequence

A geometric sequence has a constant ratio between consecutive terms.

aₙ = a₁ · r^(n-1)

a₁ = first term
r = common ratio (aₙ₊₁ / aₙ)
n = position/term number

Example: 2, 6, 18, 54, 162, ... (a₁ = 2, r = 3)

Comparing Sequence Types

Property	Arithmetic	Geometric
Pattern	Add constant	Multiply by constant
Formula	aₙ = a₁ + (n-1)d	aₙ = a₁ · r^(n-1)
Graph shape	Linear (straight line)	Exponential curve
Growth	Constant rate	Proportional rate
To find constant	Subtract consecutive terms	Divide consecutive terms

Introduction to Series

Series

A series is the sum of the terms of a sequence:

Sₙ = a₁ + a₂ + a₃ + ... + aₙ = Σᵢ₌₁ⁿ aᵢ

The symbol Σ (sigma) means "sum of."

Arithmetic Series

Sum of Arithmetic Series

The sum of the first n terms of an arithmetic sequence:

Sₙ = n/2 · (a₁ + aₙ) = n/2 · (2a₁ + (n-1)d)

In words: (number of terms) × (average of first and last terms)

Geometric Series

Sum of Finite Geometric Series

The sum of the first n terms of a geometric sequence (r ≠ 1):

Sₙ = a₁ · (1 - rⁿ)/(1 - r)

Sum of Infinite Geometric Series

If |r| < 1, the infinite geometric series converges to:

S = a₁/(1 - r)

If |r| ≥ 1, the series diverges (sum is infinite or undefined).

Sigma Notation

Reading Sigma Notation

Σᵢ₌₁ⁿ aᵢ = a₁ + a₂ + a₃ + ... + aₙ

i = index variable (counter)
1 = starting value of i
n = ending value of i
aᵢ = expression to sum

Example: Σᵢ₌₁⁵ i² = 1² + 2² + 3² + 4² + 5² = 1 + 4 + 9 + 16 + 25 = 55

Useful Summation Formulas

Sum	Formula
Σᵢ₌₁ⁿ c (constant)	c · n
Σᵢ₌₁ⁿ i	n(n+1)/2
Σᵢ₌₁ⁿ i²	n(n+1)(2n+1)/6
Σᵢ₌₁ⁿ i³	[n(n+1)/2]²

Applications

Real-World Applications

Finance: Compound interest, annuities, loan payments (geometric)
Physics: Falling objects with air resistance, bouncing balls
Biology: Population growth, bacterial reproduction
Computer Science: Algorithm analysis, data structures
Art/Architecture: Patterns, fractals, perspective

Examples

Example 1: Finding Terms of an Arithmetic Sequence

Problem: An arithmetic sequence has a₁ = 5 and d = 3. Find a₂₀ and the sum of the first 20 terms.

Solution:

Find a₂₀:

aₙ = a₁ + (n - 1)d

a₂₀ = 5 + (20 - 1)(3)

a₂₀ = 5 + 19(3) = 5 + 57 = 62

Find S₂₀:

S₂₀ = n/2 · (a₁ + aₙ) = 20/2 · (5 + 62)

S₂₀ = 10 · 67 = 670

Example 2: Geometric Sequence Application

Problem: A ball is dropped from 10 meters. Each bounce reaches 80% of the previous height. What is the total distance traveled when it stops bouncing?

Solution:

The ball falls 10m, then bounces up 8m and down 8m, up 6.4m and down 6.4m, etc.

Total distance = 10 + 2(8 + 6.4 + 5.12 + ...)

The series 8 + 6.4 + 5.12 + ... is geometric with a₁ = 8 and r = 0.8

Since |r| < 1, the infinite sum exists:

S = a₁/(1-r) = 8/(1-0.8) = 8/0.2 = 40

Total distance = 10 + 2(40) = 10 + 80 = 90 meters

Example 3: Finding the Formula for a Sequence

Problem: Find the explicit formula for: 4, 12, 36, 108, 324, ...

Solution:

Step 1: Check for arithmetic (constant difference): 12-4=8, 36-12=24 (not constant)

Step 2: Check for geometric (constant ratio): 12/4=3, 36/12=3, 108/36=3 (constant!)

Step 3: Write the formula

a₁ = 4, r = 3

aₙ = a₁ · r^(n-1) = 4 · 3^(n-1)

Verify: a₃ = 4 · 3^(3-1) = 4 · 9 = 36 ✓

Example 4: Evaluating Sigma Notation

Problem: Evaluate Σₖ₌₁⁵ (2k + 3)

Solution:

Method 1: Direct calculation

k=1: 2(1)+3 = 5

k=2: 2(2)+3 = 7

k=3: 2(3)+3 = 9

k=4: 2(4)+3 = 11

k=5: 2(5)+3 = 13

Sum = 5 + 7 + 9 + 11 + 13 = 45

Method 2: Recognize as arithmetic series

This is arithmetic with a₁=5, a₅=13, n=5

S₅ = 5/2 · (5 + 13) = 2.5 · 18 = 45

Example 5: Converging vs. Diverging Series

Problem: Determine if each infinite geometric series converges or diverges. If it converges, find the sum.

a) 1 + 1/2 + 1/4 + 1/8 + ...

b) 3 + 6 + 12 + 24 + ...

Solution:

a) a₁ = 1, r = 1/2

Since |r| = 1/2 < 1, the series converges

S = 1/(1 - 1/2) = 1/(1/2) = 2

b) a₁ = 3, r = 2

Since |r| = 2 > 1, the series diverges

The sum grows without bound (approaches infinity).

Practice

Apply your understanding of sequences and series.

1. Find the 15th term of the arithmetic sequence: 7, 12, 17, 22, ...

A) 72 B) 77 C) 82 D) 87

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

A) 48 B) 96 C) 192 D) 384

3. Find the sum: 2 + 5 + 8 + 11 + ... + 50

A) 416 B) 442 C) 468 D) 494

4. Evaluate: Σᵢ₌₁⁴ 3ⁱ

A) 40 B) 81 C) 120 D) 363

5. Find the sum of the infinite geometric series: 12 + 4 + 4/3 + ...

A) 16 B) 18 C) 20 D) 24

6. The sequence 5, x, 45 is geometric. Find x.

A) 9 B) 15 C) 20 D) 25

7. What is the common difference of the arithmetic sequence with a₁ = 8 and a₁₀ = 53?

A) 4 B) 5 C) 6 D) 7

8. Find Σₙ₌₁¹⁰⁰ 1

A) 1 B) 50 C) 100 D) 5050

9. Which infinite series converges?

A) 5 + 10 + 20 + ... B) 5 + 5 + 5 + ... C) 5 + 2.5 + 1.25 + ... D) 5 + 0 + 5 + 0 + ...

10. A geometric series has first term 16 and sum 64. Find the common ratio.

A) 1/4 B) 1/2 C) 3/4 D) 4/5

Click to reveal answers

B) 77 - a₁₅ = 7 + (14)(5) = 7 + 70 = 77
B) 96 - a₆ = 3 · 2^5 = 3 · 32 = 96
C) 468 - This is arithmetic with d=3, need n: 50=2+(n-1)3, n=17. S=17/2·(2+50)=442
C) 120 - 3 + 9 + 27 + 81 = 120
B) 18 - r = 1/3, S = 12/(1-1/3) = 12/(2/3) = 18
B) 15 - x/5 = 45/x, x² = 225, x = 15 (positive)
B) 5 - 53 = 8 + 9d, d = 45/9 = 5
C) 100 - Sum of n ones is n
C) 5 + 2.5 + 1.25 + ... - Only one with |r| < 1
C) 3/4 - 64 = 16/(1-r), 1-r = 1/4, r = 3/4

Check Your Understanding

1. How do you determine whether a sequence is arithmetic or geometric? What if it's neither?

Show answer

For arithmetic: check if consecutive terms have a constant difference (subtract). For geometric: check if consecutive terms have a constant ratio (divide). If neither difference nor ratio is constant, the sequence may follow a different pattern (quadratic, recursive, etc.) or be neither type. You might need to look for other patterns like second differences (quadratic sequences) or recursive definitions.

2. Explain why an infinite geometric series converges only when |r| < 1.

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When |r| < 1, each successive term gets smaller and smaller, approaching zero. The partial sums approach a finite limit. When |r| ≥ 1, terms don't shrink (they stay constant if |r|=1 or grow if |r|>1), so the sum keeps increasing without bound. For example, 1 + 1/2 + 1/4 + ... converges because terms shrink, while 1 + 2 + 4 + ... diverges because terms grow.

3. How is compound interest related to geometric sequences?

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With compound interest, your balance is multiplied by (1 + rate) each period, making it a geometric sequence. If you invest P dollars at rate r: After year 1: P(1+r), Year 2: P(1+r)², Year n: P(1+r)ⁿ. This is geometric with first term P and common ratio (1+r). Understanding this helps with loan calculations, investment growth, and annuities (which involve geometric series).

4. What's the connection between sequences and limits?

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A sequence can have a limit: if aₙ approaches a value L as n → ∞, we write lim(n→∞) aₙ = L. For example, 1/n → 0. For geometric sequences: if |r| < 1, then rⁿ → 0. This connects to infinite series: the sum converges if the sequence of partial sums has a limit. Understanding sequence limits is essential for calculus concepts like series convergence and the definition of the derivative.

🚀 Next Steps

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