Sequences & Series

1. The Big Idea — What Is a Sequence?

A sequence is an ordered list of numbers whose terms are generated according to a rule. The order matters: changing the order can change the sequence.

The first term is usually written as u₁, the second as u₂, and the nth term as uₙ.

Sequence vs Series

These two words are closely related but are not interchangeable.

• A sequence is a list: 2, 5, 8, 11, ...
• A series is a sum of terms: 2 + 5 + 8 + 11 + ...

How Can a Sequence Be Defined?

A sequence can be described in several ways. In examination questions, recognising which description you have been given is important.

2. Finding Patterns and the nth Term

The most important idea is that the position of a term is represented by n. If you can express the term in terms of n, you have found an explicit formula for the sequence.

Linear Patterns

Consider:

The difference between consecutive terms is constant:

This immediately suggests a linear nth-term formula.

Check:

Not Every Pattern Is Arithmetic

The differences are not constant:

But each term is multiplied by 2. This is therefore geometric rather than arithmetic.

3. Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms. This constant difference is called the common difference, written as d.

The nth-Term Formula

Why is it n − 1 rather than n?

The first term has had zero differences added to it. Therefore the nth term has had d added exactly n − 1 times.

Worked Example 1 — Basic

Find the nth term of:

Worked Example 2 — Finding a Specific Term

Find the 40th term of the arithmetic sequence:

Worked Example 3 — Finding a and d from Two Terms

An arithmetic sequence has:

Since:

Subtract the equations:

Substitute into the first equation:

4. Arithmetic Sequence Reasoning

Finding the Position of a Term

Suppose:

Which term is 64?

Does a Number Occur in the Sequence?

The algebra may produce a non-integer value. If the resulting n is not a positive integer, the number is not a term of the sequence.

Test whether 100 occurs:

Since 24.75 is not an integer, 100 is not a term.

Arithmetic Mean

If b lies exactly halfway between a and c, then:

In an arithmetic progression:

This is often useful when a missing term must be inserted between known terms.

5. Geometric Sequences

A geometric sequence has a constant ratio between consecutive non-zero terms. The common ratio is written as r.

Here:

• a = first term
• r = common ratio
• n = term number

Worked Example 1

Worked Example 2 — Fractional Ratio

Worked Example 3 — Negative Ratio

The negative ratio causes the signs to alternate.

6. Geometric Sequence Reasoning

Finding r from Two Terms

Suppose:

Dividing eliminates a:

Therefore:

Worked Example

A geometric sequence has:

Since:

Now:

Geometric Mean

If b is the geometric mean of positive numbers a and c, then:

Hence:

For example, the positive geometric mean of 4 and 25 is:

7. Recurrence Relations

A recurrence relation defines a term using one or more earlier terms.

Starting from u₁:

Arithmetic Recurrence

This recurrence generates the arithmetic sequence:

Geometric Recurrence

This generates:

Iterating a Recurrence

Consider:

8. Sigma Notation

The Greek capital letter sigma, Σ, represents summation. It provides a compact way of writing many terms.

means:

Understanding the Components

• k is the index.
• 3 is the starting value.
• 8 is the final value.
• 2k + 1 is the term being added.

Therefore:

Changing the Index Letter

The index letter is a dummy variable. For example:

The value does not change because only the name of the index has changed.

Splitting a Sigma

Constants can also be taken outside:

9. Standard Summation Results

These results are fundamental when evaluating sums involving powers of n.

Worked Example

Evaluate:

Split the sum:

Worked Example — Quadratic Expression

10. Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence.

The sum of the first n terms is:

If the final term is l:

Why Does the Formula Work?

Write the series forwards:

Reverse it:

Adding corresponding terms gives the same value every time:

Hence:

Worked Example

Find the sum of the first 30 terms of:

11. Arithmetic Series — Finding Unknowns

Finding n from a Sum

Suppose:

Use:

Factor:

Since n represents the number of terms:

Finding the First Term and Difference

If enough independent conditions are given, form simultaneous equations. Do not assume that a sequence is determined from one condition alone.

12. Geometric Series

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

For r ≠ 1:

An equivalent form is:

Both formulas give the same result.

Deriving the Formula

Start with:

Multiply by r:

Subtract:

Worked Example

13. Geometric Series with Negative or Fractional Ratios

Negative Ratio

The terms alternate in sign because r is negative.

Fractional Ratio Greater Than 1 in Magnitude

If:

then the magnitude of the terms grows as n increases.

Fractional Ratio Less Than 1 in Magnitude

If:

then the terms become progressively smaller in magnitude. This condition becomes especially important when considering an infinite geometric series.

14. Infinite Geometric Series

Consider:

If the magnitude of r is less than 1, the terms approach zero and the partial sums approach a finite limit.

In this case:

Why Does This Work?

Start with the finite formula:

If |r| < 1, then:

Therefore:

Worked Example

When Does an Infinite Geometric Series Not Converge?

If:

the infinite geometric series does not have a finite sum.

15. Infinite Series — Finding an Unknown Ratio

Suppose an infinite geometric series has first term 15 and sum 20.

The result satisfies |r| < 1, so the infinite sum is valid.

16. Applications of Geometric Series

Repeated Percentage Change

If a quantity is multiplied by a fixed factor each period, the resulting values form a geometric sequence.

For example, a machine loses 20% of its value each year. Each year it retains:

If its initial value is V:

This is geometric with:

If an infinite total of such quantities is meaningful in the model:

Repeated Bounces

Suppose a ball is dropped from height H and rebounds to 60% of its previous height each time.

The bounce heights form a geometric sequence. Questions may ask for the total upward distance, downward distance, or total distance travelled.

17. Finite vs Infinite Series

18. Connecting Sigma Notation to Series

Sigma notation can describe an arithmetic series directly.

Expanding gives:

Therefore:

Worked Example

The first term is:

The common difference is 4.

19. Manipulating Summations

Many difficult-looking sums become straightforward after separating constants and powers.

Example

Split:

Use the standard formulas:

Changing the Limits

Be careful when the lower limit is not 1.

One way to evaluate it is:

20. Working Backwards from an nth-Term Formula

Sometimes a sequence is given by a formula rather than its first few terms.

The first four terms are:

Notice that the differences are:

The first differences are not constant, so this is not arithmetic.

Second Differences

For a quadratic nth-term formula:

the second differences are constant.

21. Quadratic Sequences

Suppose:

The first differences are:

The second differences are:

Therefore try:

Since the second difference is 2A:

Therefore:

Using n = 1:

Using n = 2:

Subtract:

22. Finding Unknown Terms from Conditions

Arithmetic Example

Three consecutive terms of an arithmetic sequence are:

For consecutive arithmetic terms:

The terms are therefore:

Geometric Example

Three consecutive positive geometric terms are:

The middle term squared equals the product of the outer terms:

This condition is automatically satisfied for every non-zero x. Therefore the information given does not uniquely determine x.

23. Mixed Conditions and Simultaneous Equations

More challenging questions may combine a term condition with a sum condition.

Worked Example

An arithmetic sequence has:

From u₄:

From the sum:

From:

multiply by 2:

Subtract:

Therefore:

24. Relationships Between Terms and Sums

A useful identity for any sequence is:

Therefore:

This means that if a question gives a formula for the sum of the first n terms, you can recover the nth term by subtracting consecutive sums.

Worked Example

Suppose:

Then:

Therefore:

25. Finding a Sequence from Sₙ

The relationship:

becomes particularly useful when the question provides a polynomial expression for Sₙ.

Example

First find:

Therefore:

Check n = 1:

Also:

The result is consistent.

26. Sequences and Inequalities

Questions may ask for the first term exceeding or falling below a particular value. The process is to translate the statement into an inequality.

Arithmetic Example

For:

Find the first term greater than 100.

Since n is a positive integer:

Geometric Example

Consider:

To find when the sequence falls below 50:

This type of question may require logarithms if an exact term number is requested.

27. Understanding the Behaviour of a Sequence

Arithmetic Sequence

• If d > 0, the sequence increases.
• If d < 0, the sequence decreases.
• If d = 0, the sequence is constant.

Geometric Sequence

The behaviour depends strongly on both a and r.

• 0 < r < 1: magnitude tends to zero.
• r > 1: magnitude generally grows if a ≠ 0.
• r < 0: signs alternate.
• r = 1: constant sequence.
• r = 0: first term may be non-zero, followed by zeros.

28. Parameter Questions

A parameter is an unknown constant whose value must be determined from conditions.

Worked Example

The first three terms of an arithmetic sequence are:

Since the sequence is arithmetic:

Therefore the sequence is:

Parameter + Sum

Harder questions may then use the value of k to determine a sum or another term. The important thing is to solve the structural condition first, then continue.

29. Show-That and Proof-Style Sequence Questions

In a show-that question, the goal is not merely to obtain a numerical answer. Your algebra must establish the requested expression logically.

Example

Show that the sum of the first n odd positive integers is n².

The odd numbers form an arithmetic sequence:

Here:

Therefore:

30. Advanced Geometric Reasoning

Three Consecutive Geometric Terms

Three positive consecutive geometric terms can be represented as:

This representation is often more efficient than:

because the middle term is exactly a and the terms are symmetrically arranged around it.

Example

Three positive consecutive geometric terms have sum 26 and product 216. Let them be:

Their product is:

Therefore:

The sum condition becomes:

Divide by 2:

Multiply by r:

Both produce the same three terms in reverse order:

31. Choosing the Correct Tool

A major part of examination success is not memorising formulas independently, but recognising which structure is present.

32. Common Traps & Misconceptions

Trap 1 — Using nd Instead of (n−1)d

The nth term has n−1 jumps from the first term.

Trap 2 — Confusing Difference and Ratio

For:

the common difference is not 4. The common ratio is 2.

Trap 3 — Forgetting That n Is an Integer

A sequence term number cannot normally be 7.4 or −3.

Trap 4 — Applying S∞ Without Checking |r| < 1

The formula:

is not a universal formula for every infinite geometric expression.

Trap 5 — Sign Errors with Negative r

If r = −2, then:

Keep brackets around negative ratios when calculating powers.

Trap 6 — Assuming a Pattern Is Unique

A finite list of terms can often be fitted by many different formulas. In examination questions, the intended structure is normally signalled by additional information such as “arithmetic”, “geometric”, or a recurrence relation.

Trap 7 — Not Checking the Answer in the Original Conditions

Especially in parameter questions, substitute your result back into the condition that defined it.

33. Edexcel-Style Exam Strategy

Step 1 — Identify the Object

Ask:

• Is this a sequence or a series?
• Is it arithmetic or geometric?
• Is it finite or infinite?
• Is sigma notation involved?

Step 2 — Extract the Parameters

Step 3 — Translate Words into Mathematics

Step 4 — Check Restrictions

• Is n a positive integer?
• For S∞, is |r| < 1?
• Did a negative ratio require careful brackets?
• Does the result satisfy every condition?

Step 5 — Check the Scale

If ten positive terms are each around 100, a sum of 20 is immediately suspicious. Estimation can catch arithmetic mistakes before they cost marks.

34. Complete Sequences & Series Formula Atlas

Arithmetic Sequence

Arithmetic Series

Geometric Sequence

Finite Geometric Series

Infinite Geometric Series

Standard Sums

Sum-to-Term Relationship

Arithmetic Mean

Geometric Mean

35. Deep Worked Examples

Example A — Two Conditions in an Arithmetic Sequence

The 7th term of an arithmetic sequence is 25 and the 15th term is 57. Find the first term and common difference.

Example B — Geometric Sequence from Two Terms

A geometric sequence has u₃ = 20 and u₆ = 160. Find a and r, assuming r is positive.

Divide:

Example C — Infinite Series with a Condition

An infinite geometric series has first term 12 and sum 15. Find r.

Since:

the result is valid.

Example D — Find a Term from a Sum Formula

If:

find uₙ.

36. Very Hard Multi-Step Reasoning

Problem

The first term of a geometric sequence is positive. The sum of the first three terms is 21 and the sum to infinity is 24. Find the first term and ratio.

Let the first term be a and ratio r.

Also:

Therefore:

Substitute:

Use:

Therefore:

Check:

37. Real-World Mathematical Models

Salary / Payment Pattern

If a payment increases by a fixed amount each period, use an arithmetic model.

If it increases by a fixed percentage, use a geometric model.

Population or Growth Model

A fixed percentage growth per period produces:

Depreciation

A fixed percentage decrease gives a ratio below 1.

Repeated Physical Processes

Repeated rebounds, repeated reductions, repeated attenuation and similar processes can generate geometric sequences and series.

38. Practice Set — Foundation to Advanced

Question 1 — Arithmetic

Find the nth term of 8, 13, 18, 23, ...

Show Solution

a = 8, d = 5
uₙ = 8 + 5(n−1)
= 5n+3

Question 2 — Geometric

Find the nth term of 4, 12, 36, 108, ...

Show Solution

a = 4, r = 3
uₙ = 4×3ⁿ⁻¹

Question 3 — Arithmetic Sum

Find the sum of the first 25 terms of 2, 6, 10, 14, ...

Show Solution

a = 2, d = 4, n = 25
S₂₅ = 25/2[4+96]
= 1250

Question 4 — Finite Geometric Sum

Find the sum of the first 7 terms of 3, 6, 12, ...

Show Solution

S₇ = 3(1−2⁷)/(1−2)
= 381

Question 5 — Infinite Sum

Find the sum to infinity of 18 + 9 + 4.5 + ...

Show Solution

a = 18, r = 1/2
S∞ = 18/(1−1/2)
= 36

Question 6 — Sigma

Evaluate Σ⁸ₙ₌₁(2n+1).

Show Solution

2Σn + Σ1
= 2(36) + 8
= 80

39. Master Checklist Before the Exam

• I can distinguish a sequence from a series.
• I can identify an arithmetic sequence from a constant difference.
• I can identify a geometric sequence from a constant ratio.
• I can find the nth term of an arithmetic sequence.
• I can find the nth term of a geometric sequence.
• I can find unknown a and d from two arithmetic terms.
• I can find unknown a and r from two geometric terms.
• I can evaluate arithmetic sums.
• I can evaluate finite geometric sums.
• I can use the infinite geometric formula and check |r| < 1.
• I can manipulate sigma notation.
• I know the standard sums for 1, n, n² and n³.
• I can recover uₙ from Sₙ.
• I can solve parameter problems.
• I can handle negative and fractional common ratios.
• I can solve term-number inequalities.
• I can recognise quadratic sequences from second differences.
• I check answers against the original conditions.

That four-stage process is the core workflow for solving sequences and series problems reliably.