Module 6 Quick Reference
Sequences and Series - One-Page Cheat Sheet
Arithmetic Sequences
Definition
Add constant (d) to each term
Example: 3, 7, 11, 15, 19, ...
Common Difference
d = an - an-1
Key Formulas
nth Term: an = a1 + (n - 1)d
Sum (Series): Sn = n(a1 + an)/2
Alternative Sum: Sn = n[2a1 + (n - 1)d]/2
Sum (Series): Sn = n(a1 + an)/2
Alternative Sum: Sn = n[2a1 + (n - 1)d]/2
Quick Check: If differences between consecutive terms are constant, it is arithmetic.
Geometric Sequences
Definition
Multiply by constant (r) for each term
Example: 3, 6, 12, 24, 48, ...
Common Ratio
r = an / an-1
Key Formulas
nth Term: an = a1 · rn-1
Sum (Finite): Sn = a1(1 - rn)/(1 - r) when r ≠ 1
Sum (Infinite): S = a1/(1 - r) when |r| < 1
Sum (Finite): Sn = a1(1 - rn)/(1 - r) when r ≠ 1
Sum (Infinite): S = a1/(1 - r) when |r| < 1
Convergence: Infinite geometric series converges only when |r| < 1.
Summation Notation
Basic Notation
Σ f(k) from k=1 to n = f(1) + f(2) + f(3) + ... + f(n)
Properties
Σ(c · f(k)) = c · Σ f(k)
Σ[f(k) + g(k)] = Σ f(k) + Σ g(k)
Σ[f(k) - g(k)] = Σ f(k) - Σ g(k)
Σ[f(k) + g(k)] = Σ f(k) + Σ g(k)
Σ[f(k) - g(k)] = Σ f(k) - Σ g(k)
Common Formulas
| Sum | Formula |
|---|---|
| Σ c from k=1 to n | nc |
| Σ k from k=1 to n | n(n + 1)/2 |
| Σ k2 from k=1 to n | n(n + 1)(2n + 1)/6 |
| Σ k3 from k=1 to n | [n(n + 1)/2]2 |
Arithmetic vs. Geometric Comparison
| Feature | Arithmetic | Geometric |
|---|---|---|
| Pattern | Add d | Multiply by r |
| Find constant | Subtract terms | Divide terms |
| nth term | a1 + (n-1)d | a1 · rn-1 |
| Sum (finite) | n(a1 + an)/2 | a1(1-rn)/(1-r) |
| Sum (infinite) | Does not exist | a1/(1-r) if |r| < 1 |
| Growth | Linear | Exponential |
Application Problem Types
Arithmetic Applications
- Constant savings deposits each period
- Seating with constant increase per row
- Linear depreciation (same amount each year)
- Evenly spaced objects
- Patterns with constant addition
Geometric Applications
- Compound interest and investments
- Population growth (doubling/tripling)
- Exponential decay (half-life)
- Bouncing ball (fraction of previous height)
- Percentage depreciation (lose % each year)
Key Question: Does the problem involve repeated addition (arithmetic) or repeated multiplication (geometric)?
Step-by-Step Problem Solving
Finding a Term
- Identify sequence type
- Find a1 and d or r
- Use nth term formula
- Calculate and check
Finding a Sum
- Identify sequence type
- Find a1, an, and d or r
- Use sum formula
- Calculate total
Word Problem Strategy
- Read carefully - what is being asked?
- Identify if arithmetic or geometric
- List known values: a1, d or r, n
- Determine if you need a term or sum
- Choose the correct formula
- Solve and verify the answer makes sense
Example Template Solutions
Arithmetic Sequence Example
Problem: Find the 15th term of 7, 11, 15, 19, ...
Solution:
a1 = 7, d = 4
a15 = 7 + (15-1)(4) = 7 + 56 = 63
Solution:
a1 = 7, d = 4
a15 = 7 + (15-1)(4) = 7 + 56 = 63
Geometric Sequence Example
Problem: Find the 6th term of 3, 12, 48, 192, ...
Solution:
a1 = 3, r = 4
a6 = 3 · 45 = 3 · 1024 = 3,072
Solution:
a1 = 3, r = 4
a6 = 3 · 45 = 3 · 1024 = 3,072
Arithmetic Sum Example
Problem: Sum of first 20 terms of 5, 9, 13, 17, ...
Solution:
a1 = 5, d = 4, n = 20
a20 = 5 + 19(4) = 81
S20 = 20(5 + 81)/2 = 860
Solution:
a1 = 5, d = 4, n = 20
a20 = 5 + 19(4) = 81
S20 = 20(5 + 81)/2 = 860
Infinite Geometric Sum Example
Problem: Sum of 6 + 3 + 1.5 + 0.75 + ...
Solution:
a1 = 6, r = 0.5
Check: |0.5| < 1, so it converges
S = 6/(1 - 0.5) = 12
Solution:
a1 = 6, r = 0.5
Check: |0.5| < 1, so it converges
S = 6/(1 - 0.5) = 12
Summation Example
Problem: Evaluate Σ(2k + 3) from k=1 to 5
Solution:
= (5) + (7) + (9) + (11) + (13) = 45
Solution:
= (5) + (7) + (9) + (11) + (13) = 45
Essential Tips
- Always identify sequence type first (arithmetic or geometric)
- For arithmetic: check if differences are constant
- For geometric: check if ratios are constant
- Write down a1, d or r, and n before using formulas
- For infinite geometric series, verify |r| < 1 before finding sum
- In word problems, determine if answer is a term or a sum
- Double-check exponent calculations for geometric sequences
- Use summation formulas to save time on common sums
- Verify your answer makes sense in context
Quick Formula Summary
ARITHMETIC:
nth term: an = a1 + (n - 1)d
Sum: Sn = n(a1 + an)/2
GEOMETRIC:
nth term: an = a1 · rn-1
Finite sum: Sn = a1(1 - rn)/(1 - r)
Infinite sum: S = a1/(1 - r) when |r| < 1
SUMMATION:
Σ k from 1 to n = n(n + 1)/2
Σ k2 from 1 to n = n(n + 1)(2n + 1)/6
nth term: an = a1 + (n - 1)d
Sum: Sn = n(a1 + an)/2
GEOMETRIC:
nth term: an = a1 · rn-1
Finite sum: Sn = a1(1 - rn)/(1 - r)
Infinite sum: S = a1/(1 - r) when |r| < 1
SUMMATION:
Σ k from 1 to n = n(n + 1)/2
Σ k2 from 1 to n = n(n + 1)(2n + 1)/6