Arithmetic and Geometric Series

Review: Sequences

Arithmetic sequence: Common difference d

• aₙ = a₁ + (n-1)d
• Example: 3, 7, 11, 15, ... (d = 4)

Geometric sequence: Common ratio r

• aₙ = a₁ · r^(n-1)
• Example: 2, 6, 18, 54, ... (r = 3)

What is a Series?

Series: Sum of terms in a sequence

Notation: a₁ + a₂ + a₃ + ... + aₙ

Arithmetic series: Sum of arithmetic sequence

Geometric series: Sum of geometric sequence

Arithmetic Series Formula

Sum of first n terms:

Sₙ = n/2 · (a₁ + aₙ)

Or equivalently:

Sₙ = n/2 · [2a₁ + (n-1)d]

Where:

• n = number of terms
• a₁ = first term
• aₙ = last term
• d = common difference

Example 1: Sum First 10 Terms

Sequence: 5, 8, 11, 14, ...

Find S₁₀ (sum of first 10 terms)

First, find a₁₀:

a₁₀ = 5 + (10-1)(3)
    = 5 + 27
    = 32

Use sum formula:

S₁₀ = 10/2 · (5 + 32)
    = 5 · 37
    = 185

Answer: 185

Example 2: Using Second Formula

Find sum: 3 + 7 + 11 + ... (20 terms)

Given: a₁ = 3, d = 4, n = 20

Calculate:

S₂₀ = 20/2 · [2(3) + (20-1)(4)]
    = 10 · [6 + 76]
    = 10 · 82
    = 820

Answer: 820

Example 3: Sum of Integers

Find sum: 1 + 2 + 3 + ... + 100

Given: a₁ = 1, aₙ = 100, n = 100

Calculate:

S₁₀₀ = 100/2 · (1 + 100)
     = 50 · 101
     = 5050

Answer: 5050

Famous result! (Gauss's discovery as a child)

Geometric Series Formula

Sum of first n terms:

Sₙ = a₁ · (1 - rⁿ)/(1 - r) when r ≠ 1

Or equivalently:

Sₙ = a₁ · (rⁿ - 1)/(r - 1) when r ≠ 1

Where:

• a₁ = first term
• r = common ratio
• n = number of terms

Example 1: Sum Geometric Series

Find sum: 2 + 6 + 18 + 54 + 162

Identify: a₁ = 2, r = 3, n = 5

Calculate:

S₅ = 2 · (3⁵ - 1)/(3 - 1)
   = 2 · (243 - 1)/2
   = 2 · 242/2
   = 242

Answer: 242

Example 2: Powers of 2

Find sum: 1 + 2 + 4 + 8 + ... + 512

Identify: a₁ = 1, r = 2

Find n: 512 = 2⁹, so this is 2⁰ + 2¹ + ... + 2⁹ (10 terms)

Calculate:

S₁₀ = 1 · (2¹⁰ - 1)/(2 - 1)
    = (1024 - 1)/1
    = 1023

Answer: 1023

Example 3: Fractional Ratio

Find sum of first 6 terms: 16, 8, 4, 2, ...

Given: a₁ = 16, r = 1/2, n = 6

Calculate:

S₆ = 16 · (1 - (1/2)⁶)/(1 - 1/2)
   = 16 · (1 - 1/64)/(1/2)
   = 16 · (63/64)/(1/2)
   = 16 · (63/64) · 2
   = 16 · 63/32
   = 31.5

Answer: 31.5

Infinite Geometric Series

For |r| < 1, series converges to a finite sum:

S∞ = a₁/(1 - r)

If |r| ≥ 1, series diverges (no finite sum)

Example 1: Infinite Sum

Find sum: 1 + 1/2 + 1/4 + 1/8 + ...

Given: a₁ = 1, r = 1/2

Since |1/2| < 1, converges:

S∞ = 1/(1 - 1/2)
   = 1/(1/2)
   = 2

Answer: 2

Example 2: Repeating Decimal

Express 0.333... as fraction

Write as series:

0.3 + 0.03 + 0.003 + ...
= 3/10 + 3/100 + 3/1000 + ...

Geometric: a₁ = 3/10, r = 1/10

Sum:

S∞ = (3/10)/(1 - 1/10)
   = (3/10)/(9/10)
   = 3/9
   = 1/3

Answer: 1/3

Sigma Notation (Σ)

Summation notation: Σ (Greek letter sigma)

Format: Σ(expression) from i=start to i=end

Example: Σ(i=1 to 5) 2i = 2(1) + 2(2) + 2(3) + 2(4) + 2(5)

Example 1: Evaluate Sigma

Evaluate: Σ(i=1 to 4) (3i + 1)

Expand:

(3·1 + 1) + (3·2 + 1) + (3·3 + 1) + (3·4 + 1)
= 4 + 7 + 10 + 13
= 34

Answer: 34

Example 2: Geometric in Sigma

Evaluate: Σ(k=0 to 5) 2^k

Expand:

2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵
= 1 + 2 + 4 + 8 + 16 + 32
= 63

Or use formula: a₁ = 1, r = 2, n = 6

S = (2⁶ - 1)/(2 - 1) = 63

Writing Series in Sigma Notation

Identify pattern, write general term

Example 1: Arithmetic

Write in sigma notation: 5 + 8 + 11 + 14 + 17

Pattern: aᵢ = 3i + 2 (when i starts at 1)

Notation: Σ(i=1 to 5) (3i + 2)

Example 2: Geometric

Write: 3 + 6 + 12 + 24 + 48

Pattern: 3 · 2^(i-1) for i = 1 to 5

Notation: Σ(i=1 to 5) 3·2^(i-1)

Properties of Summation

Σ(aᵢ + bᵢ) = Σaᵢ + Σbᵢ

Σ(c · aᵢ) = c · Σaᵢ (constant multiple)

Σc = n · c (sum of constant c, n times)

Example: Using Properties

Σ(i=1 to 10) (2i + 5)

Split:

= Σ(2i) + Σ(5)
= 2·Σi + 10·5
= 2·(1+2+...+10) + 50
= 2·55 + 50
= 110 + 50
= 160

Common Summation Formulas

Σ(i=1 to n) i = n(n+1)/2

Σ(i=1 to n) i² = n(n+1)(2n+1)/6

Σ(i=1 to n) i³ = [n(n+1)/2]²

Σ(i=1 to n) c = nc

Real-World Applications

Finance: Total savings over time (compound interest)

Physics: Distance traveled with changing velocity

Business: Total revenue over multiple periods

Architecture: Amphitheater seating rows

Example: Savings Plan

Save $50 month 1, increase by $10 each month for 12 months. Total saved?

Arithmetic series: a₁ = 50, d = 10, n = 12

Last term:

a₁₂ = 50 + 11(10) = 160

Total:

S₁₂ = 12/2 · (50 + 160)
    = 6 · 210
    = 1260

Answer: $1,260

Practice