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MathematicsHigh School

Sequences & Series

Sequences and series are fundamental tools in mathematics for describing patterns, analyzing growth, and calculating sums. From compound interest to population models, they appear throughout science, finance, and everyday life.

This guide covers key definitions, arithmetic and geometric sequences, series formulas, sigma notation, convergence, worked examples, common mistakes, and a practice quiz.

1Introduction

Have you ever noticed patterns in numbers? Like the way your savings grow each month if you add a fixed amount, or how a population might double every decade? These are examples of sequences and series in action.

This topic is a fundamental part of high school mathematics, providing powerful tools to describe, analyze, and predict patterns. From finance to physics, understanding sequences and series helps us model and solve real-world problems.

Picture This

Imagine saving $50 each week. After 1 week you have $50, after 2 weeks

00, after 3 weeks
50. The amounts form a sequence (50, 100, 150, ...) and the running total is a series. After 52 weeks your savings total S₅₂ = 52/2 × (50 + 2600) = $68,900.

Real-World Uses

Finance

Compound interest, loan repayments, and annuities all rely on geometric series.

Physics

Free-falling objects (arithmetic) and radioactive decay (geometric) follow sequence patterns.

Biology

Population growth is often modeled with geometric sequences (doubling, tripling, etc.).

Computer Science

Analyzing algorithm efficiency relies on summing series and understanding growth rates.

2Key Definitions

Sequence

An ordered list of numbers. Each number is called a term. Example: 2, 4, 6, 8, ...

Term (aₙ)

An individual number in a sequence. a₁ is the first term, a₂ the second, aₙ the nth.

Arithmetic Sequence

A sequence with a constant difference d between consecutive terms. Example: 5, 8, 11, 14 (d = 3).

Geometric Sequence

A sequence with a constant ratio r between consecutive terms. Example: 2, 6, 18, 54 (r = 3).

Series (Sₙ)

The sum of the terms of a sequence. Sₙ denotes the sum of the first n terms.

Sigma Notation (Σ)

A concise way to represent a series using the Greek letter sigma for "sum."

Common Difference (d)

The constant added between consecutive terms in an arithmetic sequence: d = aₙ − aₙ₋₁.

Common Ratio (r)

The constant multiplied between consecutive terms in a geometric sequence: r = aₙ/aₙ₋₁.

Convergence

A series converges if its sum approaches a finite value as the number of terms approaches infinity.

Divergence

A series diverges if its sum does not approach a finite value (grows infinitely or oscillates).

3Arithmetic Sequences

An arithmetic sequence is like climbing stairs, taking steps of the same size each time. Each term is found by adding the common difference d to the previous term.

General Term (nth term)

aₙ = a₁ + (n−1)d

a₁ = first term, n = term number, d = common difference

Worked Example

Find the 10th term of: 3, 7, 11, 15, ...

a₁ = 3, d = 7 − 3 = 4 (identify values)

a₁₀ = 3 + (10−1) × 4 (plug into formula)

= 3 + 9 × 4 = 3 + 36

a₁₀ = 39

Tip

To find d, subtract any term from the next: d = aₙ₊₁ − aₙ. If the differences aren't constant, it's not arithmetic!

4Geometric Sequences

A geometric sequence is like a chain reaction, where each step multiplies the previous one by a constant ratio r.

General Term (nth term)

aₙ = a₁ × rⁿ⁻₁

a₁ = first term, n = term number, r = common ratio

Worked Example

Find the 5th term of: 2, 6, 18, ...

a₁ = 2, r = 6/2 = 3 (identify values)

a₅ = 2 × 3⁴ (plug into formula)

= 2 × 81

a₅ = 162

Tip

To find r, divide any term by the previous one: r = aₙ₊₁/aₙ. If the ratios aren't constant, it's not geometric!

5Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence. There are two equivalent formulas depending on what you know.

When aₙ is known

Sₙ = n/2 (a₁ + aₙ)

Average of first and last term, times the number of terms.

When d is known

Sₙ = n/2 (2a₁ + (n−1)d)

Derived by substituting aₙ = a₁ + (n−1)d into the first formula.

Worked Example

Find the sum of the first 10 terms of: 3, 7, 11, 15, ...

a₁ = 3, d = 4, n = 10 (identify)

a₁₀ = 3 + 9 × 4 = 39 (find last term)

S₁₀ = 10/2 × (3 + 39) (sum formula)

= 5 × 42

S₁₀ = 210

Interactive Calculator

Enter values to find the nth term and sum of the first n terms.

6Geometric Series

A geometric series is the sum of the terms in a geometric sequence. There is a separate formula for finite and infinite series.

Finite (n terms)

Sₙ = a₁(1−rⁿ) / (1−r)

Valid when r ≠ 1.

Infinite (|r| < 1)

S∞ = a₁ / (1−r)

Only converges when |r| < 1. If |r| ≥ 1, the series diverges.

Worked Example 1: Finite

Find the sum of the first 5 terms of: 2, 6, 18, ...

a₁ = 2, r = 3, n = 5

S₅ = 2(1−3⁵) / (1−3)

= 2(1−243) / (−2)

= 2(−242) / (−2)

S₅ = 242

Worked Example 2: Infinite

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

a₁ = 1, r = 1/2

|r| = 1/2 < 1 → converges

S∞ = 1 / (1 − 1/2) = 1 / (1/2)

S∞ = 2

Convergence Check

Always check |r| < 1 before using the infinite sum formula. If |r| ≥ 1, the series diverges and there is no finite sum.

7Sigma Notation (Σ)

Sigma notation provides a compact way to write sums using the Greek letter Σ (sigma), which stands for "sum."

Σ f(k), k = m to n

k = index variable, m = lower limit, n = upper limit, f(k) = expression for each term

How to Read Sigma Notation

  1. Start at the lower limit: Substitute k = m into f(k) for the first term.
  2. Increment k by 1 and substitute again for the second term.
  3. Continue until k reaches the upper limit n.
  4. Add all resulting terms together.

Worked Example

Evaluate Σ(3i − 1) from i = 1 to 4

i=1: 3(1) − 1 = 2

i=2: 3(2) − 1 = 5

i=3: 3(3) − 1 = 8

i=4: 3(4) − 1 = 11

Sum = 2 + 5 + 8 + 11 = 26

Notice this is an arithmetic series with a₁ = 2 and d = 3.

Tip

When you see sigma notation, first write out a few terms to identify whether the series is arithmetic, geometric, or something else. Then apply the appropriate sum formula.

8Worked Examples

Example 1: Theatre Seats (Arithmetic)

A theatre has 20 seats in the first row, 22 in the second, 24 in the third, and so on. How many seats are in the 15th row?

a₁ = 20, d = 2, n = 15

a₁₅ = 20 + (15−1) × 2

= 20 + 28

a₁₅ = 48 seats

Example 2: Total Theatre Seats (Arithmetic Series)

What is the total number of seats in the first 15 rows?

a₁ = 20, a₁₅ = 48, n = 15

S₁₅ = 15/2 × (20 + 48)

= 7.5 × 68

S₁₅ = 510 seats

Example 3: Bacteria Growth (Geometric)

A bacteria culture doubles every hour. If there are 100 bacteria initially, how many after 5 hours?

a₁ = 100, r = 2, n = 6 (at 0h, 1h, ..., 5h = 6 terms)

a₆ = 100 × 2⁵

= 100 × 32

a₆ = 3200 bacteria

Example 4: Investment (Geometric Series)

$500 invested at the start of each year at 5% interest compounded annually. Total after 4 years?

Each investment grows: 500(1.05)⁴ + 500(1.05)³ + 500(1.05)² + 500(1.05)¹

a₁ = 500(1.05) = 525, r = 1.05, n = 4

S₄ = 525(1.05⁴ − 1) / (1.05 − 1)

= 525(0.21551) / 0.05

S₄ ≈

263.81

Example 5: Infinite Geometric Series in Sigma Notation

Evaluate Σ 10(1/5)ᵈ⁻₁ from k = 1 to ∞

a₁ = 10(1/5)⁰ = 10, r = 1/5

|r| = 1/5 < 1 → converges

S∞ = 10 / (1 − 1/5) = 10 / (4/5)

S∞ = 12.5

9Common Mistakes

Confusing arithmetic and geometric

Always check: is there a common difference (add/subtract) or a common ratio (multiply/divide)? Misidentifying the type leads to using the wrong formula entirely.

Using n instead of (n−1)

In aₙ = a₁ + (n−1)d and aₙ = a₁ × rⁿ⁻₁, the exponent/multiplier is (n−1), not n. The first term has 0 differences/ratios applied.

Mixing up sum formulas

There are separate formulas for arithmetic series, finite geometric series, and infinite geometric series. Double-check you're using the right one for the problem type.

Forgetting the |r| < 1 condition

An infinite geometric series only converges when |r| < 1. If |r| ≥ 1, the series diverges and the infinite sum formula cannot be used.

Errors with negative ratios and exponents

Be careful with signs: (−2)³ = −8 but (−2)⁴ = 16. The sign alternates with odd/even powers when the common ratio is negative.

Quick Revision Summary

  • ✓A sequence is an ordered list of numbers; a series is their sum.
  • ✓Arithmetic: constant difference d. General term: aₙ = a₁ + (n−1)d.
  • ✓Geometric: constant ratio r. General term: aₙ = a₁ × rⁿ⁻₁.
  • ✓Arithmetic series: Sₙ = n/2(a₁ + aₙ) or Sₙ = n/2(2a₁ + (n−1)d).
  • ✓Geometric series: Sₙ = a₁(1−rⁿ)/(1−r) for finite; S∞ = a₁/(1−r) for |r| < 1.
  • ✓Sigma notation (Σ) is shorthand for writing series compactly.
  • ✓An infinite geometric series converges only when |r| < 1.
  • ✓Always identify the sequence type first before choosing a formula.

Frequently Asked Questions

What is the difference between a sequence and a series?
A sequence is an ordered list of numbers (e.g., 2, 4, 6, 8). A series is the sum of those numbers (e.g., 2 + 4 + 6 + 8 = 20). Think of a sequence as the terms listed out, and a series as what you get when you add them together.
How do I know if a sequence is arithmetic or geometric?
Arithmetic: check if the difference between consecutive terms is constant (a₂ − a₁ = a₃ − a₂ = ...). Geometric: check if the ratio between consecutive terms is constant (a₂/a₁ = a₃/a₂ = ...). If neither is constant, it may be a different type of sequence.
Can the common difference or common ratio be negative?
Yes! If d is negative, the arithmetic sequence decreases (e.g., 10, 7, 4, 1, ...). If r is negative, the geometric sequence alternates in sign (e.g., 2, −4, 8, −16, ...). Both are perfectly valid.
When should I use the aₙ formula versus the Sₙ formula?
Use the aₙ (nth term) formula when you need to find a specific term in the sequence. Use the Sₙ (sum) formula when you need the total of the first n terms. Identify what the problem is asking for — a single term or a sum.
Are sequences and series used in real life?
Absolutely! Compound interest and loan payments use geometric series. Simple interest uses arithmetic series. Population growth is often geometric. Free-falling objects follow arithmetic patterns. Algorithm efficiency in computer science relies on series analysis.

Practice Quiz

Test your understanding of sequences and series — select the correct answer for each question.

1.What is the common difference in the sequence: 5, 9, 13, 17, ...?

2.Find the 5th term of the arithmetic sequence: 2, 5, 8, 11, ...

3.What is the common ratio in the geometric sequence: 3, 6, 12, 24, ...?

4.What is the sum of the arithmetic series 1 + 2 + 3 + ... + 100?

5.An infinite geometric series converges when:

6.In a geometric sequence with a₁ = 5 and r = 2, find a₄.

7.Evaluate Σ(3n) from n = 1 to 4.

8.What type of sequence is 1, 4, 9, 16, ...?

9.Find the sum of the infinite geometric series: 1 + 1/3 + 1/9 + ...

10.In the Fibonacci sequence F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂, find F₆.

Final Study Advice

  • 1.Always identify arithmetic vs geometric first -- it determines which formula to use.
  • 2.Write out the first few terms to verify your formula gives the correct sequence.
  • 3.For geometric series, always check |r| before deciding on a finite or infinite formula.
  • 4.Practice expanding sigma notation by hand -- it builds intuition for recognizing patterns.
  • 5.Use the interactive calculator above to experiment with different values and check your work.

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